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.
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".
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.
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.
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.