The upcoming Carnival of Citizens sports a "Church and State" theme. There's a lot that could be said here, but for this post I want to look at how religion fits with a deliberative conception of democracy.
The right policy is that which the general public would converge upon, following informed ideal deliberation. In reality, that ideal remains out of reach; but we do the best we can. As a deliberative democrat, I hold that we should promote informed deliberation among citizens, in hopes that the best-justified positions will ultimately carry the day.
Receptivity is a key value here: it's vital to note that public debate is not merely another instrument of power, manipulating others to do as you want. Rather, it is seen as a co-operative, rational enterprise. We all have the shared goal of promoting justice and the good. We may have divergent ideas about what exactly this involves. But, recognizing our own fallibility, we remain open to the possibility of changing our minds, if faced with stronger opposing arguments. In short, deliberative democracy is about civic respect, or the commitment to an inclusive and collaborative politics: working together to discern the right action, rather than unilaterally forcing my views on others.
So, where does religion fit into all this? I guess that depends on the nature of the religion, and the way one tries to bring it into politics. If one's religion is based on public reason, then I see no problem in principle. For example, if you think that God's hand is evident in nature, and his perfect character transparent to reason, then you may try to bring me to see this. If there are good reasons to think that scripture provides an accurate moral guide, then you can share those reasons with me. We might argue about the correct interpretation, or even about whether the purported Holy Book is a relevant guide at all, but those are issues to be settled through deliberation; the answers are not "given", or something we can know prior to inquiry. They are entirely appropriate for public debate.
On the other hand, the more dogmatic forms of religion have no place here, for they are inconsistent with civic respect. For example, if you are certain that the truth has been revealed to your group alone, and that all others are irredeemably blind to it, then you will be incapable of meaningful deliberation with them. The dogmatist is not receptive to alternative possibilities, and may see no point in collaborating with "morally degenerate" infidels. He has no respect for their civic autonomy -- their ability to reason about what sort of society ours should be, or how we should live together. To the dogmatist, sure of his own infallibility, people who disagree are merely obstacles to achieving what he already knows is "right".
There's nothing essentially religious about such attitudes, of course. Dogmatism comes in all stripes. But religion can be especially conducive to it, and the risk is heightened if a religious group happen to form a majority. They may seek simply to impose their private reasons ("faith") on the broader public, without any adequate justification. But that's not democracy. It's civil war.