Love and compassion are arguably universally accepted by all religions and secular ethical systems. Not murdering or torturing people randomly is also probably highly uncontested by everyone.
However, what would be examples where religious moral reasoning and secular moral reasoning part ways with significant disagreements in the ethical sphere, why is this the case, and is there any hope that these disagreements might eventually be resolved based on reason and evidence?
The main discrimination between religious and secular moral reasoning is
Broadly speaking, religious reasoning declares as right what God as the master law-giver has declared as right, and wrong to be the oppposite. While secular reasoning does not invoke a theistic law-giver. Instead it considers ethics as a convention to reach certain social goals, e.g., peace, justice, stability.
The resulting ethics must not necessarily be different. Because religions often just establish socially successful rules as divine commands, see many of the Ten Commands from the Jewish religion.
An especially prominent one right now is the moral status of living as a member of LGBTQ+. The negative view of people in this group spans several religions and is justified by an appeal to some flavor of natural law: it's not the way we are "meant" to behave. It is often backed up by reference to scriptures and doctrine that can (usually very loosely) support the view that God disapproves of LBGTQ+ behavior.
From a secular ethical point of view, it's hard to see why we should be especially worried about this group nor how their being and acting out their life is intrinsically a threat to others.
To be fair -- Marx/Engels and the communists were not much better here (or at least were uneven). Their reasons were also ideological or a false association (e.g., between homosexual behavior and pedophilia)
As a first step you could simply go through the Ten Commandments. Some of them are consistent with secular morality:
Others are specific to Judeo-Christian religion
Religions are human efforts to recognize and value what is sacred. Secular ethics or morality either recognizes and values some things as sacred, and therefore shares primary attributes with religion; or it desecrates everything, it does not hold anything to be sacred, and can therefore distinguish itself from religion.
What are examples of moral principles in religions that secular ethical systems find hard to accept or justify and why?
The most obvious example is that religions tend to discriminate strongly against non-believers. Famously, quite a few religions made it pretty clear that it is acceptable (or even beneficial) to kill, or at least heavily penalize every human that did not profess to their creed. See the crusades in the middle ages, or modern-day military jihad for some pretty strong examples.
Aside from that, equally obvious, by having their religious background lore, any moral principle based on that is also, by force, different from secular ethical systems, and very hard to accept. For example, any moral that is based around professing your belief to the deity/deities or following arguably random religious rites, or things like Indulgence, can only be had in a religious framework, not in a secular one.
Love and compassion are arguably universally accepted by all religions and secular ethical systems. Not murdering or torturing people randomly is also probably highly uncontested by everyone.
This seems highly opinionated and cynical to me. Love, compassion, not murdering or torturing only is uncontested as long as the "people" also belong to the same religion, or unless you redefine significantly what those terms mean.
Some religions did torturing (or arguably at least murdering) to their own people just fine, i.e. Aztecs. If you were unlucky enough to live during the time of the Spanish Inquisition, a mere word from your neighbour could get you tortured and murdered. If you were a retainer to some high noble, your answer as to whether Ancient Egyptian retainer sacrifices was murder or not was probably different from the nobles' ones.
Yes, you could argue that these atrocities were not due to "moral principles in religions", but just misguided power structures acting under the umbrella of a religion, but that's like saying that the Pope is not Christianity. For a great many people, he definitely is, and they will take on morals that the Pope prescribes just fine - after all, it is one of the primary function of religions to hand moral rules to their people.
Then there are religions like ancient Norse or Greek mythology which are filled to the brim with deities waging wars amongst themselves, killing humans left and right (i.e. using "their" humans as armies to throw against each other), lying and stealing. I am not an expert, but I sense very little "love and compassion" in those old texts (mostly compartmentalized to few individual deities whose job it was to "do" love and compassion, like Freyja and Sigyn), it's almost like that is just not a topic that was very relevant in those times (when basic survival was much more important to most people).
Of course this is not to say that secular societies didn't do just the same, I'll spare myself the effort to find links to some pretty wild examples in the 20th and 21th century. But these are the examples I can come up with that differ between religious and secular ethical systems the most, as asked.
My perspective, for what it's worth, is that religious "morality" is really a mixed bag of a number of concepts, including (but not limited to):
I imagine secular ethics will struggle with all but the first concept, because no matter how convoluted the ethical argument, at the end, it is expected to arrive at the identification of harm done to other.
There aren't such examples, really. At least, not in a universal, logical sense; in a statistical sense, there are certainly ideas that a majority of religious people alive today would hold and that a majority of secular people would disagree with.
The fundamental issue is that the question casts a very broad net by contrasting religious reasoning in general with secular reasoning in general. The question even acknowledges this breadth by referring to secular ethical systems in plural: the ethics associated with democracy, communism, capitalism, ethnic supremacy, libertarianism, anarchism and so forth can a priori all be secular, despite their vast differences. That is to say, secular reasoning is simply a term that refers to any reasoning that is not religious; as such, it can justify anything that one pleases, as long as it does not require belief in divinities, the supernatural, and so forth.
For instance, perhaps a religion says that people should not eat pork because God has so decreed it. That is a religious argument. But one can imagine any number of secular arguments for the same: people should not eat pork because all living creatures have value, because pigs are anatomically very similar to humans, because pork production is bad for the environment, because pigs have too many parasites, because red meat is bad for one's health, or because pork is just disgusting, man. These arguments need not even be correct—they are still secular, and there are people who believe all of them.
In fact, there's evidence to suggest that a large proportion of religious prescriptions and proscriptions are little more than the secular perceived common sense of their original milieu, codified and given a divine mandate in order to encourage compliance.
Less obviously, secular reasoning can even justify most arguments that do directly relate to religious matters (e.g. "people should worship deities") through arguments like "This religion is obviously false, but people should follow its dictates because it connects them with their cultural history/it will help me attain political power/it promotes societal cohesion/most people here already believe it, so why rock the boat." These are purely secular arguments that do not rely on the truth of any religion, and yes, people have made all of them.
In short, the answer is that secular reasoning, writ large, can justify essentially anything that religious reasoning can. There can be a secular argument for anything that a religion promotes, albeit that any secular arguments for intrinsically religious strictures like "people should believe in God" will very likely be somewhat cynical ones.
Regarding the part about whether the disagreements "can be resolved based on reason and evidence," I would have to say, sometimes. Reason and evidence can resolve disagreements on facts, so in theory (for instance) someone might be convinced to became an atheist (or believe in a deity) on the basis on evidence. Contra Searle, I do not think they can usually resolve fundamental differences in moral axioms, though they may convince someone that they have reasoned incorrectly from them; for instance, "You believe that God's most fundamental commandment is loving your neighbor but you also think that it's fine to hate them because their skin color is different" might convince someone if the first part is actually their belief.
That said, in some cases, it may not be necessary for secular ethical systems and religious ethical systems to agree—is it necessarily an issue for most secular ethical systems that Jews and Muslims think it is wrong to eat pork, as long as the latter are not forcing the secular reasoner not to eat it? On the other hand, in some cases—at least from my ethical perspective, and from many others—certain religious beliefs might be too negative to leave unchallenged, such as the aforementioned "God wants me to hate my neighbor because of their skin color"; unfortunately, people are likely to either not be convinced to change their beliefs at all, or if they are persuaded, to be persuaded by appeals to implicit moral beliefs that are actually more important to them than the bigotry that they justify with religion (e.g. "I realized that group X are actually nice people, and I can't really hate anyone who is nice to me.")
However, what would be examples where religious moral reasoning and secular moral reasoning part ways with significant disagreements in the ethical sphere
Everything where social and cultural norms have evolved from the time the religious norms were established. Two examples would be slavery and punishment. For example, in the holy books of all three Abrahamic religions, it is clear that slavery and the death penalty for even relatively minor crimes is perfectly normal.
Note that the biblical norms are not enforced anymore in most of Western society. They can still be found in parts of the Islamic world. And even in the USA and Europe, you need to go back only a few hundred years to find them alive and well.
why is this the case,
Because religion has trouble changing its basic statements, e.g. its holy texts, because doing so invalidates the claim that they are god-given. Why would an all-mighty, all-knowing god give people rules on slavery when slavery itself is wrong? Why didn't he tell people 2000 years ago that it is wrong?
Secular morals have no such issue adapting. On the contrary, changing cultural norms are an accepted fact and the morals 500 years in the future will very likely differ in at least some aspect from today.
and is there any hope that these disagreements might eventually be resolved based on reason and evidence?
No. At least as you ask. Because "reason and evidence" are not accepted by religion as valid reasons to change holy scripture.
I do believe that they will eventually be resolved, but in a different way. Most likely another reform movement within the religion that finds a way to reconcile a considerable update to the moral codex with the holy texts.
A (mostly) trivial example would be the ethics of one's relationship with divine beings. If the secular perspective is one in which such beings are not in view, then a secular ethics might not be expected to say anything about such relationships. On the other hand, Locke for example held that we have a natural sense of obligation in terms of our having been created by his one God, and the principle of obligation as dependent on creaturehood might seem secular, if by "secular" we mean "not scriptural, but empirical and abstract." But if we anthropomorphize our deities enough, we'll want our relationships with them to not be grounded in abstract guesses about their intentions, but we will want a "revelation" of those.
So another trivial example would be the ethics of scripturation: most (major) religions need their adherents to be very invested in this or that specific book, with all that such an investment demands (such as learning ancient Greek and spending inordinate amounts of time arguing with people online about ancient Greek grammar and semantics: a great good, the scripturalist thinks, and a waste of time, the secularist should think).
Kant said that self-discipline is freedom, the reason is that people have free will, can rely on free will to get rid of natural instincts, restrain themselves from doing those things that they think they should not do.
Religion, in any sense, is essentially an extension of a natural instinct. All the rules of fraternity and mutual aid are in essence ultimately directed to the ultimate goal of the natural instinct: the maximum continuity of the race. It does not need to reason, union, mutual aid is the natural conditions for life to be better continued, and it has long been rooted in our genes.
So, in this sense, although both religious and secular ethical theories require people to do good, the starting point is completely different. Religion explains the demands of nature; philosophy explains the demands of freedom.
One religious principle is to believe that one's views are necessarily correct and that there is a need to proselytize. Secular thinking tends along the lines of, "I was somewhat correct today, and hopefully more correct tomorrow." So no one much feels a need to crow about being normal, sometimes getting things right, and intending to improve.
On the other hand, humans don't come with good reasoning or behavior built in (obviously), so there is certainly a need for education. The teachers I had in school often exuded a skeptical attitude bordering on cynicism, so they seemed the farthest thing from 'promoting' any sort of view or attitude towards secular society.
However, what would be examples where religious moral reasoning and secular moral reasoning part ways
There are two very important aspects to this question which I feel are not sufficiently covered in other answers.
First is the difference between stated moral and actual actions. History is ripe with examples of individuals, organizations, countries even giving word to following a specific religious moral, but in reality doing something completely different. Saying one thing and doing something totally different is not unknown.
As a side note: It can sometimes be traced back to a moral standpoint about the difference between "us" and "them", in essence giving a moral ground for as example persecution. Historic examples include christians persecuting jews and various other minorities. Classical examples includes the handling of black persons, both in slave trade and by example Ku-Klux-Clan (which consider themselves to be both religious and having high morals). Modern examples are not too difficult to find either.
The second aspect is historical context, as hinted above. Secular moral reasoning is a comparatively modern koncept. Before that basically only different religious moral concepts existed. As witnessed some were quite different from the religous concepts today. In addition over time most religions has changed their moral concepts so what we see today is often quite different from older interpretations. We should not expect the morals of today in, say the roman catholic church, to be the same as in older times of, say the early christian church only open to jews.
So to answer part of the question: several religious moral reasonings start in dividing people into "us" and "them". Most secular reasoning starts from the basic tenet that all persons are worth the same.
The answer by @Groovy has been deleted because its rhetoric has been assessed as “a bit too harsh to stand as-is, unnecessarily so”. IMO the following statements from Groovy's text deserve a rational discussion on a philosophical site like StackExchange Philosophy:
Let's take a cold, hard look at the facts, shall we? Every major religion has scripture and dogma that, if followed to the letter in this modern age, would have us living as barbarians. The "good book" advocating for the subjugation of women as mere property, the stoning of nonconformists, the murder of infidels, and a total disregard for individual liberty. This isn't morality - it's a text-book case of immoral savagery.
And you want examples? How about the Judeo-Christian tradition that considers the human species so wretched and debased that we require a heavenly dictator's daily supervision and the looming threat of eternal torture? There's a wonderful moral framework for healthy human development. Or Islam mandating death for apostasy and blasphemy? So much for the Enlightenment values of free thought and expression that any rational ethical system is built upon. Perhaps the Hindu caste system's spiritual pedestal for oppression and discrimination based on an accident of birth?
The examples could fill tomes, but you get the idea. Religion claims a monopoly on morality, while having the gall to enshrine the principles of our darkest, most tribalistic impulses as divine command. Secular ethics, once disentangled from the shackles of faith, at least holds the promise of evolving towards kindness, rights, and reason. Any rift between the two worldviews isn't some mere disagreement to be resolved through patient discourse. It is an ideological battle between enlightenment and obfuscation. One path leads towards wisdom and human flourishing, while the other is a regression to the wretchedness that kept our species mired in the muck until so very recently.
So count me out of this quest to find common ground with the immorality, intolerance and anti-rational dogma of religious "moralities." Our secular ethical systems may not be perfect, but they at least hold the hope of growing, adapting and progressing. The other? Well, it's trapped in the prejudices of the nomadic iron-age goatherders who conceived it.
