Do they get to pop a top after a great victory, or do they simply have a quiet dinner at home with their wife(s)?
Since no one has actually really answered the OP yet, I'll give it a shot...
There is no requirement that players or coaches at BYU must be Mormon. But they must adhere to signed code of conduct that conforms to the beliefs of the Mormon Church, which means no alcohol, tobacco, etc. This tends to discourage a lot of non-Mormons from playing for/coaching at BYU. There have been a few notable exceptions. Jim McMahon -- an Irish Catholic -- famously flaunted the school's honor code during his days at the school, which is why you never hear his name mentioned among BYU fans as one of their all-time greats (even though he is arguably the best QB they've ever had). Additionally, another non-Mormon, Mohammed Elowonibi, won the Outland Trophy for BYU in 1989, but as an observant Muslim, he had little trouble conforming to the school's honor code.
All that said, because BYU is operated by the Mormon Church, there's a strong Mormon culture that surrounds it and its sports teams. While the head coach doesn't have to be Mormon, it goes over much better with the fans and boosters when the coach is "one of them." Therefore, to my knowledge, all of BYU head football coaches have actually been Mormon.
As for Utah, it's a completely different story. As the secular, state-run school, it tends to be the team that most non-Mormons in Utah root for (althought the Utes have plenty of Mormon fans, too). It also happens to be in Salt Lake City, which has the largest concentration of non-Mormons in the state. Thus, although plenty of Mormons have coached and played at Utah, being Mormon is absolutely not a requirement -- and definitely not culturally expected like it is at BYU.