One must remember that the First Amendment prohibits any law respecting an Establishment of Religion or prohibiting the Free Exercise thereof. Although there are many debates about it; the issue becomes very clouded over issues such as state funding for parochial schools. Is it proper for students at church schools to be transported on buses which are paid for by taxpayer money? Should students at such schools receive textbooks furnished by the state? At what point has one crossed the line?
Other problems exist due to the conservatism of many religious groups. When George W. Bush ran for President, he delivered a speech at Bob Jones University in Greenville, S.C. which is ultra conservative. He received a great deal of criticism for this. At the same time, religion and politics are not necessarily a fatal mixture. Our currency still reads "In God We Trust," our Pledge of Allegiance still contains the words "Under God," and Congress opens each session with a prayer led by a Chaplain.
The main problem that religion contributes to in today's American politics is the tendency that we have to become very polarized as a country. Religion plays into the "culture wars" and helps to make Americans see each other as very different people.
There are major differences between the attitudes of liberals and conservatives when it comes to religion. Many liberals feel that the conservatives are trying to impose their religious values on the country, making it a theocracy. At the same time, conservatives often think that liberals are trying to drive God out of the public sphere and impose athiesm on the country.
Because these are issues of faith and religion, they are hard to compromise on and it is easy to demonize the other side. This means that arguments about this kind of issue drive Americans further apart, making them think the other side is not just wrong, but actually in some way evil.