Maybe a very tiny minority of people actually believe that BYU should win games because of the church affiliation. In fact, I don't think I have ever met anyone that really believed that.
Do some BYU fans express a a moral highground over fellow members of the church that root for Utah? I probably agree with that. There have been plenty of posts on this board that would probably back that up ("how can you root against a team that represents the church's university" comes to mind). A lot of BYU fans also assume that Utah students were not qualified to attend BYU and have no problem expressing that opinion on this board. This would really annoy me if I attended Utah.
However, they definitely don't think that BYU wins because of church affiliation. That's just not true.