Deliberate mistake: Father Merrin says, "It is He who commands you", and we hear and see the ceiling crack above his head. In the next shot we see the ceiling above him and they are no cracks in it, but four shots later we see the exact same spot and now cracks appear even though we hear no more cracking over-dubs after the first one.
Deliberate mistake: Det. Kinderman walks up the stairs where Burke Dennings fell to his death and looks up at Regan's window. This could not possibly be the outside of her window because when we see the room from the inside as Chris closes the window near the beginning of the movie, there's at least 6 to 8 feet between the window and corner of the house. Now when we see it as he looks up, the window is right at the corner of the house.