Which philosophers have argued that a God cannot exist and why? What are some of the major positions involved, and perhaps some common responses?
Your question as reformulated is very broad; people write whole novels on these subjects. I will try to cover the most famous arguments, organizing my responses in terms of the argument first, with references afterward to a person or people who posited the argument, followed by common responses.
The Problem of Evil
Perhaps the most famous argument against the idea of God is the Problem of Evil, which targets specifically the Abrahamic god (the god of Judaism, Christianity, and Islam who is considered to be omnibenevolent, omniscient, and omnipotent). Various versions of the Problem of Evil has been espoused by numerous philosophers over the ages, going back several centuries. Most notable of these include Epicurus, Hume, Leibniz, Kant, Cousin, Kreeft, and Hatcher. The argument generally goes as follows:
- God exists.
- God is omnipotent, omniscient, and perfectly good.
- A perfectly good being would want to prevent all evils.
- An omniscient being knows every way in which evils can come into existence.
- An omnipotent being, who knows every way in which an evil can come into existence, has the power to prevent that evil from coming into existence.
- A being who knows every way in which an evil can come into existence, who is able to prevent that evil from coming into existence, and who wants to do so, would prevent the existence of that evil.
- If there exists an omnipotent, omniscient, and perfectly good being, then no evil exists.
- Evil exists (logical contradiction).
A well-known refutation to the argument is Alvin Plantinga's The Free Will Defense, which essentially says that God allows evil in order to allow humans to have free will, which is a greater good). Consider also reading John Hicks refutation, The "Soul-Making" Theodicy, and Fyodor Dostoevsky's The Brothers Karamazov which also touches upon this subject in a story-like format.
The Ultimate Boeing 747
"The name comes from Fred Hoyle's amusing image of the Boeing 747 and the scrapyard. ... Hoyle said that the probability of life originating on Earth is no greater than the chance that a hurricane, sweeping through a scrapyard, would have the luck to assemble a Boeing 747. ... This, in a nutshell, is the creationist's favorite argument - an argument that could be made only by somebody who doesn't understand the first thing about natural selection ..." –Richard Dawkins, The God Delusion, pg. 113
In reality, it is the Design Argument in reverse, which itself comes from the Argument from Improbability. The Argument from Improbability states that complex things could not have come about by chance, thus, because many things in nature seem complex, people extend the idea to use God as a designer. However, as Dawkins points out, "come about by chance" is not equivalent to "come about in the absence of deliberate design". Evolution by natural selection offers a solution that has nothing to do with chance and does not need to invoke a God. But what the Ultimate Boeing 747 argument also argues is that, as a method of explaining complexity, we often invoke the idea of a designer to explain how a complex object came about. For example, a beautiful painting or an amazing sports car - these things did not just "happen" by chance, they were designed and created. But in the case of the the argument from design, proponents posit that God did all the designing, an entity which would be even more unlikely to have come about by chance than a hurricane assembling a Boeing 747. God is the Ultimate Boeing 747.
Argument from Free Will
The Argument from Free Will is the idea that omniscience and free will are incompatible (technically, free will is also incompatible with omnipotence as well, since omnipotence encompasses omniscience). The basic idea is that if God knows what you are going to do in the future, that means your future is determined, which removes any possibility of free will. If you are truly free, not even God would have the ability to predict what choices you could make.
This argument has been used by numerous philosopher over the ages, and as a generic refutation my Philosophy of Religion books don't even ascribe a particular author. Counter-arguments include altering the conception of free will to prevent a contradiction, such as the idea that free will is merely freedom from coercion, not necessarily freedom from causation. Also, opponents of this argument has tried to reformulate the idea of omnipotence—the Wikipedia article on this topic does a decent job of describing the counter-arguments here, most notably those from Thomas Aquinas and C. S. Lewis.
Another contradiction which goes against the existence of God is the Omnipotence paradox, which essentially claims that the very idea of omnipotence is a paradox, therefore God as typically formulated can't logically exist. In simple form it is often expressed as a question: Can an omnipotent being create a stone so heavy that it cannot lift it? Again, here the Wikipedia article provides a surprisingly thorough analysis of the issue, so I'm not going to belabor the point here.
For a general introduction to the philosophy of religion and the major arguments over the ages against (and for) the idea of God, I would recommend Philosophy of Religion: An Introduction and the companion book Philosophy of Religion: Selected Readings by William Rowe. It offers a comprehensive look at the classic arguments going back many centuries up until modern times. For more modern arguments (and popular readings), I would recommend first and foremost Sam Harris' books, particularly Letter to a Christian Nation and The End of Faith. Richard Dawkin's also wrote a few books on the subject as well, the most notable being The God Delusion, although it gets dry at parts in the book. Christopher Hitchens also wrote an excellent novel called god is not Great: How Religion Poisons Everything that you might be interested in.
Rebecca Goldstein has published a novel entitled 36 Arguments for the Existence of God which contains an appendix listing 36 philosophical arguments for the existence of God, along with the flaws found in each argument. Her husband, Steven Pinker, is also a well-known atheist, and has published on the subject as well. I'd recommend her list as a good starting point for the commmon arguments in both directions.
On a much more sophisticated level, Martin Hägglund argues (somewhat controversially, but persuasively, in my opinion) that Derrida's thought is based upon a Radical Atheism.
And, of course, all Buddhist philosophy is predicated on the notion that there is no all-powerful, creator God (although minor, relatively ineffectual deities are permitted.)
EDIT: As this answer has started drawing downvotes, I thought I'd flesh it out a bit more. My answer above was based on what I thought the spririt of the question was, and not the actual letter of the question. I will now treat of the actually stated question.
I see that the question asks about arguing that "a God" cannot exist. We need to be clear here, because there is a significant difference between arguing that a particular god cannot exist, and that no god can exist-- the latter argument is impossible to make, unless we have a clearly agreed-upon set of properties for any possible god, which is extremely unlikely.
So, the question then becomes, Which philosophers have argued that a particular God cannot exist?
Now, at this point, we must turn to the question of modality. The question, as stated, talks about the argument that a particular God cannot exist, rather than does not exist. This means that we are looking for arguments that the non-existence of particular God is a necessary fact, and merely a contingent fact.
Arguments of this sort are therefore going to depend upon the notion of logical impossibility.
One type of argument will be that the properties attributed to the particular God are logically inconsistent. For example, one can argue that certain notions of omnipotence are self-defeating, and therefore it is impossible that any God possessing this notion of omnipotence necessarily cannot exist. Bertrand Russell takes this tack in parts of Why I Am Not A Christian where he argues for the impossibility of a God who is defined as a First Cause. One also finds arguments of this type in the classical/medieval Indian debates between the Buddhists and the Nyāya regarding the existence of the God Īśvara.
Another, slightly more complex argument, is going to depend upon a conflict between the properties attributed to the God, and the actually existing state of affairs in the world-- for example, arguments of Theodicy. This is a more problematic case, depending on the framework of the the question; there is the danger that we come back to arguments of contingency --the argument that the presence of evil in the world demonstrates that God does not exist, but if we take the presence of evil to be contingent and not necessary, we are then left with the possibility that although God does not exist, there could have been a perfect world with no evil and a God.
However: one should not limit oneself to atheist positions, if one is attempting to answer the question as written-- most religious arguments for the existence of a particular God contain arguments against the existence of other Gods. So, many traditional arguments for the existence of the Christian God include (or imply) an argument that that god Thor cannot exist, as these arguments stipulate as a property of the Christian God that He is the only god.
I'm responding here chiefly to the final point of Ben's of the merits of atheism versus agnosticism, because I think the core issue he has here is perhaps not being addressed in the edited question.
Strong atheism, in the sense of maintaining that it is absolutely true that no god exists, is not a very popular position, and hard to justify logically. Maintaining that no god can exist is stronger still, and still less popular, and so the answers you get to the question you asked will be few.
Weak atheism does not assert that no god exists, but merely refrains from asserting any god exists. This is a much more popular position, as it is easily defensible from within the epistemology of science, which draws heavily on Popperian falsificationism. In scientific statistics, the null hypothesis - that the things measured are not linked by any relationship - is assumed true as a starting position, and invalidating it requires showing that the empirical data gathered have a low probability of being encountered if the null hypothesis holds. At no point is the null hypothesis ever claimed to be absolutely true; it is merely held to be the only rational default position until evidence shows otherwise. To tie this back to atheism, those using this line of thinking to reason about supernatural entities begin with the default assumption that no such entity exists, and will require proof before they budge from it. Dawkins, who is one examplar of this philosophy, calls this "Tooth Fairy agnosticism" - he cannot prove that there is no such thing as a Tooth Fairy, so his belief that there is no Tooth Fairy is not absolute, but he nonetheless evaluates it as exceedingly unlikely that such a being exists.
Arguments against the particular gods espoused by particular religions exist, of course, mostly exploiting inconsistencies of mythology or morality, but I gather those are not what you're looking for here.
The atheists are the philosophers arguing against the existence of god(s).
Among this list of people some are inevitable: Nietsche, Russel, Camus.
I would also cite other well known atheists (but not really philosophers): Linus Pauling, Alan Turing and Richard Dawkins.
There was also atheists in antiquity; Leucippus and Democritus who first imagined the atomic theory of matter.
Our ancestors mainly needed supernatural explanations to account for bizarre empirical phenomena (thunders, earthquakes, but also the sun's heat, or why it does rain) or to answer ontological questions (why do I exist, where did the Earth comes from).
Following that, most of the atheist arguments are that as we understand more and more deeply the World, there is less and less need for god(s) to explain our Universe.
Common responses are, yes we know increasingly about our Universe, but we don't (and we won't) understand 100% of it, hence, god is still needed to explain the rest (for instance in Evolution vs Creationism, god is still needed to account for the lack of some transitional fossils).