I think priests should have some 'mystical' effects, but they should be so subtle and slight that the player can't really be sure that they're there at all. For instance, undead might just shy away from a priest from time to time, and even demons might slow down a little bit as they charge one. Not enough to actually
save them, of course, but the effect would be neat.
Later, if hauntings and so forth were brought in, a priest could help avert and dispel them. If a poltergeist took up residence in one of your rooms, a priest could run in to do the old bell-book-candle routine.
Priests could, in general, serve more as the player's way of fighting back a little against magic and magical creatures, rather than a way of using it themselves.
...oh, they would also have to attend to weddings and funerals, of course. That goes without saying. Having a skilled priest attend to a funeral might help reduce the unhappy thoughts associated with the death, the same way it helps people come to terms with it in the real world.
And they'd collect tithes. I would imagine any sort of dwarven religion would be big on tithes.
And mandate the creation of religious junk and the establishment of more temples. Although they might let you sell some of that junk if they think it'll earn them converts.
I don't think we need to be too specific about the religion's details... although for some reason when I think of dwarves, I'd expect their religion to be akin to Judism. Maybe it has to do with the beards. Lots of big sacred scrolls covered with religious text in an ancient language would fit dwarves, too. And Tolkien also had them mourning for their lost homeland.
Perhaps the noble who covers religion would just be a sort of high priest, and the player would have to assign other dwarves as lesser priests under them. When the high priest arrives, they'd bring with them a big sacred scroll which would form the center of the main temple the same way an anvil is brought in for your first forge; if anything happens to that scroll (like, say, water damage), every religious dwarf in your fortress would probably freak out. The death of the high priest would send them all into mourning, too.