As I understand it, exorcism and demonic possession are part of the faith of the Catholic church and of certain fundamentalist sects. For most protestants, to say nothing of the pantheon of other religions out there, it's a bit of an alien notion. Regardless, it seems that people from all stripes enjoy exploring the idea of possessions. The Rite does a good job in this respect; for the most part, the religious aspects of the film were quite interesting and well done.
The protagonist of the film is a young priest who is also an atheist. It's hard to believe the character because he maintains his atheism even after witnessing all kinds of demonic phenomenon. In the same situation, I would be running to church to sing hallelujah.
The tactics of the demon are also questionable. It's not giving away anything to say that the antagonist spends the duration of the film essentially training the protagonist to exploit his only weakness. For being an all-power hell beast, he is pretty stupid.
The film succeeds in creating a spooky environment, and garners a few cheap and not-so-cheap scares. Unfortunately it's failings are more than skin deep, and the viewer is left wishing things had been a bit different.