Much as I would like to believe that most therapists can leave large parts of their belief systems outside the door, so to speak, I think where there are differences such as the one's you suggest Solarwind, especially if the T has for some reason revealed her/his views in some way, there is cause for caution.

I think it comes down to the issue of trust you seem to also be concerned with. I, for example, would have trouble trusting a 'very religious' T partly because I don't trust any 'very religious' people, and partly because I would consider it a boundary violation for my T to talk to me about their religion or lack thereof, unless I asked a direct question about it. If my T were a vegan, on the other hand, I would only find that problematic if there was some kind of judgement of my food choices or pressure to change as a result of that. That would quickly become a deal breaker for me.

I have had the experience of being treated by a T whose world view lead her to espouse and practice some of the more esoteric, but quite popular methods in circulation today. I did try some of them, for compliance sake and because I do try to give 'different' things a chance. But in the end I left her because she kept returning to these methods even after I had said that they didn't really work for me. (A very polite way of saying what i really thought ) So, she wasn't actively trying to persuade me of anything, but still, her belief system affected how she did therapy, and that eventually made us a bad match.
I guess the only thing I can really say is, you are the one who is supposed to be in control of what happens in therapy. That doesn't mean not opening up and letting yourself be known and helped, but it means that you are allowed to say no to anything that does not feel right to you, even if it is something you might later change your mind about.

As someone said above, therapy is about building a relationship. We do not have to choose to stay with anyone as our therapist who we would not normally be comfortable relating to in some other type of relationship, be it a friend, or even a 'close acquaintance'. Bottom line, if it makes you uncomfortable, say so. If it stays uncomfortable, and interferes with the relationship building process, look elsewhere.