Well,
If he was a Sunni or Shi'a muslim (as opposed to a Nation of Islam muslim, which I don't think would fit the OP), he would not have been associated with that nutball pastor in Chicago - so he would have avoided some political difficulties at that point. However, it seems from some reports that his participation in the congregation was instrumental in his earlier (local/state) political career - so we must find something else to serve as his connection to his voting base.
Assuming that can be butterflied away somehow, I think he might actually do better during the 2008 campaign - and might have come to prominence earlier. Presuming he handles the earlier prominence adroitly...
1. Folks who would vote against him for being muslim, were (and did in OTL) going to vote against him for other reasons anyway - so nothing lost there.
2. He might actually pick up some muslim/immigrant votes in places like Virginia, California, and the Great Lakes - (I think he won those anyway - so no EV impact) - which wouldn't change the tally, but by making those EV more secure, would free the campaign to spend more time/money elsewhere.
About the only negative might be issues with the jewish portion of his donor base. The question would be - what is more important to said donors - their political beliefs, or their ethnic-religious ties to Israel (and the level of 'threat' they perceive from a muslim Obama)? I think it would cut in favor of politics over the other ties, so I think the impact to his funding would be minimal, or perhaps even a net plus based on more donations from muslims.
I can't see anything in his policies that would appreciably change.
Lastly, being muslim and publicly so (I don't see Obama as the type to dodge on questions of his religious beliefs), it would defang that portion of vitriol against him.
"He's a closet Muslim!"
"Bwuh? He's been perfectly open about that since day one. Be nice to have someone in the White House who wasn't drunk or chasing skirts half the time."
Of course, he would probably have to take an even tougher line on man-caused-emergencies and contingency operations, at least rhetorically.