In the basement scene where Father Karras visits the Georgetown home rented by Chris, the 2x10 ceiling joists are fairly new looking despite the fact that it is a Federal-style home (c. 1780-1830) revealing it as a built set and not an actual location of that era.
William Peter Blatty closely modeled the exorcism scene on the actual rite of exorcism in the Church's "Rituale Romanum". Father Merrin can be seen opening a copy of the Rituale in the scene in question. However, the priests depart from the Rituale in two important details. First, there should have been four people (apart from Regan) in the room during the exorcism: the exorcist himself; an assistant priest to take over in case the exorcist died midway through; a member of the victim's family of the same sex as the victim, to help restrain her; and a doctor, to (among other things) administer any medication that was needed. Due to the "2 Priest rule", Fr Merrin should have delayed the second round of the exorcism and phoned the bishop to get a replacement for Fr Karras, instead of trying to tackle it on his own.
In the 2000 re-release, when Father Merrin asks Chris McNeil what Regan's middle name is, she is wearing a fur coat. When the priests enter Regan's room and Father Karras closes the door in her face, Mrs. McNeil is no longer wearing the coat. This was fixed on some versions of the DVD by adding a coat digitally.
At one point in the exorcism, Karras stands right at the end of the bed as it begins to thump on the floor. There's a cut to a wider shot showing this end lifting off the floor but Karras has vanished. A subsequent shot shows him still standing in that spot.
When father Karras is in the dorm sitting on the bed smoking a cigarette, the priest takes the cigarette from Karras and lays him down on the bed. When father Karras sits up and grabs the priest's arm, the cigarette is still in Karras's right hand. In the next shot, the cigarette is back with the other priest.
In the basement with the Ouija board, Regan is playing with a table tennis paddle and ball. She puts the ball under the paddle and rests it on the table when her mother brings the Ouija board. Clearly the ball is under the paddle as evidenced by the angle of the paddle. In the very next shot, the paddle is flat on the table and the ball has disappeared. In the next shot, the ball reappears as demonstrated by the angle of the paddle.
After the initial session of exorcism, Father Damien Karras unties Regan's wrist from the bed and ties them together. When Father Merrin returns to resume the session Regan's wrists are again tied to the bed.
When Father Karras first goes to visit Regan, one of the bedside tables next to her bed is gone, along with the rest of the extraneous furniture from the room; later, this bedside table is back in place, though with a different lamp than was initially on it.
Despite Dennings being killed the police do not close off the roads or even investigate the McNeil household. If a person is found dead outside a house or other building like in the film the police would have to search the place for clues.
Steaming hot water comes out of the tap as quickly as Damien turns on the faucet. It's not possible for water that hot to come through plumbing that fast, especially since the bedroom was on the highest floor of the house and the temperature of the room was at least freezing.
The first medical test Regan has is an arteriogram. This test does require contrast dye to be injected into a large artery, but it is usually the femoral artery (groin) or brachial artery (arm). Very rarely would the carotid artery in the neck be used. The femoral artery is most often used based on its easy access.
During the exorcism, Regan is lying on the bed with her head to one side while green vomit is pouring from her mouth onto the bed. When she turns her head to the other side, the tube that pumps this "vomit" is visible on the back of her cheek, just under her ear, which in turn is shown devoid of make-up after the pressure from the pump tube lifts the prosthetic mask away from her neck.
When Karras first leaves the McNeil household we see Kindermann from his car looking at a person behind the curtains which is implied to be Regan despite the fact she was strapped to her bed. That's the point of the scene: That Kinderman sees what should not be possible: Regan moving freely about despite being ill and strapped to the bed.