Rav Miller zatzal said that Jews should stay out of politics.
There are a few other issues:
Shabbos would be a major issue (Joe Lieberman wrote a sefer about Shabbos and the issues that he has had as a senator and a Vice Presidential candidate).
shaking hands with women – at rallies and the like (Each handshake with a woman would be a separate safek issur d’oraysa for which one is possibly required to let himself be burned alive rather than to transgress).
A frum president would be required to oppose things that are against the Torah, like homosexuality.