I don't tend to talk about religion/politics/controversial things etc. much and honestly have no idea of the viewpoints friends or even many family members/relatives have on many of those subjects. No real reason why... they just don't come up.
With that said, I think that the only way certain beliefs would cause any kind of an issue is if I felt that the person were pushing their beliefs on me or were judging me because my beliefs may be different from their's and they therefore think I'm 'wrong' and view me differently for it and we didn't have enough commonalities to balance out the differences.
I feel that I am a pretty easy going and open minded person and get along well with the majority of people and really don't tend care a lot what somebody's beliefs are when it comes to most major issues, unless they're way out there or are something that just does not mesh well with how I feel at all. Like in Linds' description of that discriminatory woman badmouthing somebody transgender, something like that would not fly well with me. I strongly disagree with discrimination of any kind and would have a very, very hard time talking to a person with such viewpoints unless I had to (like if they were a customer of mine or something - I'd keep it business only, no friend option).
I do however think that it is easier to feel closer to and have a better relationship sometimes with those that have similar viewpoints on most things. Sometimes opposite is good too, but it's nice to share common ground on different things. Makes me feel a bit more comfortable sometimes, just so I don't say something offensive or something by accident.
I did have a friend growing up who was super religious. One of my best friends. However, on occassion I found certain topics difficult to discuss, because I am not religious at all and there was a moment that I felt she thought that was weird or wrong of me, but it was a fleeting moment and I think just an insecurity on my part. But for as religious as she was, the topic didn't come up that much and when it did, while sometimes I didn't know what to say, it didn't really bother me either. We shared plenty enough other things in common. So no, I've never ended a friendship/had one end based on opposing view points. While I don't entirely know the viewpoints on a variety of subjects that the people I know have, the ones I do know of that differ to mine have never been enough to be relationship ending.