In the original Director's cut of the film, the name of the demonologist recommended by the psychic is Dr. Johan Abrams. In the Theatrical version, every time his name is mentioned it is noticeably re-dubbed as "Dr. Johan Averies".
(at around 51 mins) Katie's hair changes lengths from one scene to the next. This is especially noticeable when Katie tells Micah he is finished using the Ouija board and her hair is long, then in the next scene she is on the couch and her hair is much shorter.
(at around 1h 14 mins) Katie's weight fluctuates noticeably from scene to scene. This is especially noticeable when she is looking at the computer about the other possessed woman. From one shot to the next, she is much heavier.
A little over 2 minutes into the movie, Micah is making dinner. There is clearly a bottle of red (spaghetti?) sauce and a take out coffee cup on the counter. Katie pours them both a glass of wine. Next cut his wine glass is gone, she is drinking a bottle of water and on the counter is a bag of lettuce or some other veggie (not part of their dinner) and two bottles (vitamins). Then they're eating dinner and she has her wine glass back.
(at around 1h 19 mins) When Katie is saying she doesn't want to leave the house, it is assumed that Micah has set the camera on the bed, as it stops moving and stays at a constant angle. This explains why the camera stayed while Micah walked away.
(at around 18 mins) Towards the beginning of the movie, when Micah and Katie are talking with the psychologist, there is a hard cut from the doctor to Katie as the doctor is talking, but the dialogue does not skip. Since the movie is supposed to be entirely self-filmed footage, this would imply that either there were two cameras filming at once, or that they did a second "take" with the doctor, both of which are unlikely.