When you say, “priest” are you speaking of a Roman Catholic clergy, specifically? Or also including Protestant minsters? And are you including Jewish Rabbis?
Regardless, as an Pagan atheist and Recovering Catholic, I would think that a minister, priest, or Rabbi wold probably be better at his job and more effective in helping his congregation if he really WAS a True Believer. That is, believed in a personal, Theist God. A biblical type, that is, which listens to and answers our prayers.
That being said, I am 100% positive that there are scores of minsters, priests and rabbis out there who do NOT believe in a Theist type of God. And for a variety of reasons: some may have once been believers, but their Faith has waned—as it has for many of the once-religious types. This is very natural, since we all know that there has never been the slightest, scantest evidence of god existing. Of ANY sort.
And some in the priesthood and the ministry never believed, and get into their occupation for different reasons. Power, control, manipulation, et al. And we don’t even have to “go there” with the pedophile issue and the Roman Catholic Church.
The fact that we have in the past decade or so discovered the RCC to be absolutely infested with pederast priests surprises me not the least. I have been aware of this for years, personally. And also am aware that the ministry and the priesthood attract a good deal of sociopaths. Indeed, when you think about it is the perfect profession for one, due to its vast opportunities to garner ill-placed trust and then take advantage.
And, even as a very anti-religion atheist, I find this a shame.
I will close by saying that, even though at the outset I said I thought it would be beneficial for a priest to believe in God, I DO think that a good man, who really wants to help people, could still be a good priest or minister even if he knew the truth: that there is virtually no way a biblical, theist God could exist. For, he could be like a counselor, a confidant, and a motivator. Remember that one does not have to believe in God to learn from and follow the teachings of Jesus. He doesn’t have to believe that Jesus was God, or even was in any way divine. So. a non-believing priest could incorporate Jesus’ teachings into his helping of others.
I personally feel that by far, Jesus’ teachings are the best and most useful part of the Bible.