Visible crew/equipment: When Marion is dead, you can see the camera panning back reflected in her eyes.
Deliberate mistake: When the CA. trooper is at the front of her car, the Arizona license plate is shown. In 1989, Arizona began issuing a single license plate and that plate is always on the back of the vehicle. This is deliberate, as the remake follows the original movie shot by shot, and there the trooper checks the front license plate.
Continuity mistake: Marion gets into the bathtub/shower, and unwraps a bar of soap, throwing the wrapper down into the tub. Later when Norman is cleaning the bathtub, the wrapper has vanished.
Continuity mistake: When Marion searches for a place to hide the money, the lamp on the nightstand is over to the left side. However, when Marion places the newspaper on the nightstand, the lamp has moved to the center.
Revealing mistake: As a homage to the original; when Norman carries Marion's dead body, wrapped in the shower curtain to the car, you can see her bra strap through the curtain.
Deliberate mistake: The car that Marion buys has a current California license plate with the number NFB 418, which matches the number from the original movie - but at the time, California plates would have had another digit before the letters; probably a 3, i.e. 3NFB418.
Continuity mistake: When Norman is spying on Marion through the holes in the wall, the long shot shows Norman's face about a foot away from the smaller hole. In the closeup, his eyeball is right up against the smaller hole on the outer sheetrock wall. When we return to the long shot, he's back to a foot away.
Continuity mistake: When Norman shows Marion her room, the bed that is reflected in the mirror (that Norman walks past) doesn't have a nightstand next to it. However, when Norman and Marion stand in front of the mirror and talk, there is now a nightstand next to the bed.
Character mistake: When Marion is calculating how much she spent her math is wrong. The movie shows 400000 - 4036 = 395963. It should be 395964.
Factual error: At the end of the shower murder, the camera pans back from Marion's eye, one of the most iconic scenes in cinema. If we are to presume she is now dead however, the nature of muscles surrounding her pupil would cause them to relax and her pupil would dilate, not go to a pinpoint as depicted.