As someone who has 4 alts that have done Torghast now, I can say that shadow priests do ok in Torghast. They do ok at killing and have pretty good anima powers, and self healing obvs helps to hang in there if it gets rough. The main problem is that it can be fairly slow and excruciatingly boring.
As a tank spec, Torghast is an absolute joke - my vengeance DH can move through a level in under a minute if I get the right anima powers. With a priest, I'd be lucky to get to what, the 2nd room in that time? Clear power is also radically different - again, with the same DH I can pull the whole floor to the final boss and explode them at once, whereas with a priest I have to take it slow and careful unless I get lucky with powers. This is probably common to all non-pet casters though, since we're heavily disadvantaged by not being able to instantly cast or cast on the move.