In a continuing effort to shorten my list in the sidebar despite constantly adding to them, I'm removing my two posts on the same God issue with Judaism, Christianity, and Islam and linking just this one that refers to both of them.
The first was simply an argument that the one being called God in English is the same being that Muslims refer to with the name 'Allah', though they believe very different things about this one God, which makes all the difference. The second looked at some arguments in N.T. Wright's The New Testament and the People of God about first-century Judaism and first-century Christianity and their relevance to this issue. Partly it helped clarify my position, and partly it helped me express my reasons for thinking this a little more clearly.
Update: This comments on this post about on freedom, rights, and 'under God' in the pledge of allegiance also has a discussion about the same issue that reflects a development of my thought on this issue. The basic idea is that I don't think there's one sense in which an expression might refer to God. An act of false worship may in one sense be worship of God but done so wrongly that it's immoral and worthless. The same act of false worship may in another sense not count as genuine worship of God and therefore count as worship of a false god. I think the former sense is primary and the latter secondary, as the terms in English are standardly used, and John 8:41-44 and II Kings 17 contain examples of the two senses in the Bible. The correct theory of reference-fixing for terms like 'God' and 'Allah' should explain why there are both senses, but it may still turn out that my view that the one of II Kings 17 is the primary one in English.
Update 2 (March 2008): The conversation picked up again in response to a debate between Rick Love and John Piper. In Muslims Worshiping God But Not Worshiping God, I present the argument in a different enough way that I thought it was worth linking to it here, and I respond to the objection that Muslims deny an essential property of God and thus must not refer to him when they use God-language.
Then in Is the Father of Jesus the God of Muhammad? I respond to a couple arguments from Timothy Tennent. He argues that it doesn't seem right to say that the God of Muhammad is the Father of Jesus, and I point out the real problem in that statement is that no Muslim or Christian should accept both labels for God, but that both could refer to God in the same way that mistaken descriptions of mere human beings can refer even if they get things badly wrong (e.g. the red-haired man across the room drinking champagne, when it's a bald woman in a wig cross-dressing and drinking wine out of a champagne glass).
Finally, in Islam and a Different Jesus, I respond to a more difficult set of arguments from Kevin Courter and Dale Tuggy. What about Paul's statement in II Corinthians that those who teach a false gospel are teaching a different Jesus? What about his statements in I Corinthians about demons lying behind idols? What if Muhammad actually received the Qur'an from a demon? This post offers a slightly modified view that can handle these objections, and it argues that, since you won't be able to modify the alternative view to handle the difficulties I've raised for that position, my view is the most reasonable way to handle the tensions within the scriptures that bear on this issue.
Update 3 (March 2011): Miroslav Volf has now released a book about this that seems worth reading. He sees to me to give a bad argument for the correct view, and then he goes on to apply the view in ways that strike me as possibly going way too far, but I'd have to see the details of what he has to say to be sure beyond that. I've recorded some thoughts on Volf here.