The most adverse effect religion has on US politics is the 'values voters' segment of the population who vote based on issues that are morally out of date as opposed to voting based on facts and broader policy concerns. The right basically gets to use the religious populace as a hate group that will vote for them because they funnel hatred and ignorance to court the votes.
Beyond elections, there is a ridiculous imperative to support Israel based on religious and political backlash potential.
Yes, I hate it when the bible thumper get up on their high horse and say the bible says this, (of course they ignore all the parts of the bible that they don't want to follow).
It makes me wonder if people even read the bible, the bible is fine with slavery, and in fact you can sell you daughter into slavery and the bible is OK with it. Exodus 21:7, the killing of the first born Exodus 11:5, wow that's a kind and loving god, kill someone who can't even make the law of the land, or a killing babies, killing non-believers, witches, etc.
I know they say that's the old testament, but Jesus said he was there not to replace the laws of his father, but fulfill them so the old testament in still in force.
I know we'll never get religion out of politics, their voting block is too big, unless you happen to be Muslim of course, or Jewish, their voting block is really too small for them to have much influence except on the local level. And when a non-Christian group does start to gain power on the local level, you should hear the Christians scream. They are more than happy to point out the flaw in any religion except their own, that holds true for most followers of any religion, they are blind to the failing of their religion, while take great offense at the flaws of another. And that's the big reason to keep religion out of politics, it creates tension when we should be working toward an agreement that is fair to everyone.
Take the abortion issue, there are plenty of religions that are fine with it, so religious objections should be discounted, and the question should be at what point if any should abortion be illegal? After all if you don't want an abortion don't have one, who are YOU to say what is best for another person?
Yes, but there is a very strong proviso to this.
As Reza Aslan points out, religion isn't just one's sect or one's church or whether or not a person reads their holy book. Religion is an identity. So when we want people to get religion out of politics, all too often we are asking for people not to have democracy.
Still, the way that we allow religion to influence politics is flawed. We allow it to affect public spaces that should be neutral religiously, like posting the Ten Commandment in courthouses (though that picture has improved). We mix in God with money and the Pledge of Allegiance, which is a bit like mixing Skittles and ranch dressing in that it ruins them both.
People should vote based off of their culture, but they should not vote against other cultures. White people in the United States should certainly vote by their values, but they should not vote as a group to try to maintain their power over others. (And, yes, people of color should vote only for what would be actually fair, not what would happen to enrich them).
We need a culture of citizenship, and institutions that project that citizenship. That idea has to say, "I am a Christian, but I vote as an American".