Why do Republicans act like a president being Muslim would be an impeachable offense?

You mean that small, unrepresentative chunk of Republicans? Those ones that most others wish would go away and stop embarring the party? Those ones? I ume because US relations with the Muslim world are hostile, and having a Muslim president is not something they're comfortable with. I don't think a whole lot of people would be comfortable with it, actually, whatever they said to appear tolerant, only Obama's not a Muslim, so it's irrelevant anyway.