There are numerous cases where the views held by different people clash. Homosexual intercourses are one such obvious example: many religious denominations (for example Catholicism) consider them sinful, while many other people uphold that there is nothing wrong with such intercourses.
In cases when two people with mutually incompatible views wish to cooperate or coexist without constant arguments (or one of them getting the boot) they may both agree to avoid uttering views the other one finds unacceptable. For example, a Catholic may agree to avoid vocally condemning homosexual sex, knowing that a Liberal believes that such views cause real, tangible and severe harm to homosexual people (and is willing to protect such people from this perceived act of aggression and discrimination by banning the offenders from online communities).
It would seem that this can only work if the non-negotiables of both parties are negative ("don't do X", for example "don't say it is wrong to be homosexual"), rather than positive ("you must do X").
The gender pronoun issue seems to be one such example that frustrates the above. This is because the non-negotiable¹ in many places such as online communities, workplaces, universities seem to go beyond the negative rule, that is, "do not label others with a gender that does not match their stated gender identity". Rather, the non-negotiable seems to be positive: "you must use pronouns that acknowledge other's stated gender identity"².
At the same time it can be argued that the teaching of certain Christian denominations (such as the Catholic Church) is that a person whose biological sex is, say, female, remains female even if such a person has a male gender identity³. It can be therefore argued that if, for example, a person's preferred gender pronoun is he/him and it is known that this differs from this person's biological sex then using this pronoun positively reaffirms this person's identity as a male, which automatically rejects this part of the teaching of such religions.
Since (it seems to me) a Catholic (or other religious person whose faith has similar teachings on this subject) cannot abide by the rule to use preferred gender pronouns without verbally rejecting the teachings of their religion, the rule that requires the use of preferred gender pronouns (as opposed to a weaker, non-positive version of such a rule) effectively requires all such people to reject their faith in order to participate. This is equivalent to forbidding all Catholics (and many other religious people) from participating, even though such an outcome does not seem to be intentional⁴.
Does the requirement to use preferred gender pronouns preclude the participation of people whose faith teaches that gender cannot be changed in communities that implement this rule? (The problem seems to be important since more and more places are implementing such rules.)
¹ "In the Code of Conduct, we state “We don’t tolerate any language likely to offend or alienate people based on race, gender, sexual orientation, or religion — and those are just a few examples.” To clarify, this includes discussion questioning or debating the legitimacy of someone’s race, gender, sexual orientation or religion. Further, the debate or discussion of whether or not you want to or should use a person’s pronouns is both offensive and alienating."
² "The Code of Conduct has two direct references to pronouns: • “Use stated pronouns (when known).” • “Prefer gender-neutral language when uncertain.”" This means that, if one's preferred pronouns are known, their use is compulsory (instead of gender-neutral language). To a certain degree the issue can be circumvented: "You can often avoid using pronouns altogether. It's actually pretty rare to need third-person pronouns at all on most Stack Exchange sites." Even so, circumstances may arise when this positive requirement can no longer be avoided: "But conspicuously avoiding using pronouns for one group of people while using them normally for others is a way of refusing to recognize their identity, and that is discriminatory. Please don’t do that."
³ "In 2015, the Vatican declared that transgender Catholics cannot become godparents, stating in response to a transgender man's query that transgender status "reveals in a public way an attitude opposite to the moral imperative of solving the problem of sexual identity according to the truth of one's own sexuality" and that, "[t]herefore it is evident that this person does not possess the requirement of leading a life according to the faith and in the position of godfather and is therefore unable to be admitted to the position of godfather or godmother." In June 2019, the Catholic Church published a document titled "Male and Female He Created Them", which summarized its official position. The document rejected the terms transgender and intersex, and criticized the idea that people could choose or change their gender as a "confused concept of freedom" and "momentary desires". It asserted male and female genitalia were designed for procreation."
⁴ "I sincerely disagree that transgender people are who and what they say they are. Am I still welcome on Stack Exchange sites? You are, but some of the things you might want to say aren't."