Continuity mistake: When Bond escapes from the Willard Whyte facility in the desert in the moon buggy, a car that is chasing him goes over a ridge and turns on its side, as the buggy speeds by. A wheel rolls back into the shot, all on its own. It doesn't belong to the car or the three-wheelers also chasing, but looks like it belongs to the buggy, which has its full complement of wheels all through the scene.
Plot hole: While it is essential to the story, there is a problem with apes in the space ship. While it is the one that took Taylor to the future, there is no way the chimps could have salvaged the ship from the water (plus there wasn't enough time for such a salvage operation. Kira and Cornelius were living in the ape city we see in "Beneath the Planet of the Apes" never mentioning to Brent that they found a spaceship he could travel back with). They didn't have the technology for such an operation, no diving suits, and chimps are scared of water. Also, the water would have destroyed the power systems of the ship when it sank. (00:00:25)
Visible crew/equipment: Very early on in the movie, Neville is happily driving around the deserted streets. Half way through this scene the camera shows us Neville's view out the front windscreen of the car. Ahead, moving cars can be seen crossing the intersection, obviously where the streets hadn't been blocked off for the filming of the movie.
Continuity mistake: On Day 2, when the two officers go into the town's doctor's office, they leave the van with the driver's door way open, beyond 90 degrees, almost 180. The door didn't move to indicate it corrected itself as they went in. When they come out, the door is almost closed. And there has been no indication of a wind other than the helicopter, and that is supposed to be 1000 feet up and not even close to the office.
Visible crew/equipment: When LUH 3417 closes the door of the medicine cabinet because she doesn't have the right to take more pills, there's a reflection of a crew member's hand retaining a door.
Visible crew/equipment: In the lower left-hand pane of glass in the kitchen cupboards, a crew member can be seen moving a rat (Ben) from the center table to where Willard left the box of rat poison. (01:30:42)
Factual error: When Mission Control tracks the orbiting Apollo spacecraft, the ship's ground track up on the big map has it moving across North America parallel to a line of latitude. With the exception of orbiting directly over the Equator, this is impossible.