It seems to me Steve Chapman at Reason Online has a good point (as did Atrios and Mark Kleiman) about "Mitt Romney's strange double standards":

Romney said there should be no religious test for this office. "A person should not be elected because of his faith nor should he be rejected because of his faith," he said.

Rejected because of his faith, no. But rejected for his lack of faith? That's another question. Romney evinces a powerful aversion to skeptics. "We need to have a person of faith lead the country," he said in February, which sounds like a religious test to me.

Now one can surely argue that a person's beliefs on religious issues are relevant to our evaluation of the person's character — of his commitment to reason, or his likely adherence to a moral code, or whatever else. (Our commenters had some very interesting thoughts on some closely related issues here and here.) Such relevance is in my experience often overestimated, but there's nothing inherently unsound about such arguments. [Sentence initially inadvertently omitted:] Likewise, one can reasonable argue that on balance one's beliefs on theological issues are in practice far enough removed from one's likely behavior on secular matters -- and that their significance is so likely to be erroneously estimated by others who have different theological beliefs -- that it's better to just exclude them from political discussion.

But it seems odd to say that challenging Romney because of his Mormon faith — for instance, "How can you believe such odd-seeming factual assertions about reality and history?," or "Your religion barred blacks from full-fledged membership until 1978, when you were 30; what was your view at the time about the morality of this prohibition?" — is somehow categorically improper, but rejecting atheist or agnostic political leaders because of their lack of religious faith would be permissible.

I realize, by the way, that Romney's "We need to have a person of faith" line was an apparently off-the-cuff response to a hostile questioner. But I haven't heard of Romney expressly or implicitly disavowing it, despite the public attention the line had gotten. Nor have I heard of anything else that suggests that Romney does not in fact take this view.