(at around 14 mins) During the intelligence test that Lewis conducts on Zira with the blocks, the position on the blocks shifts between shots, even when Lewis isn't jumbling them.
The ship the apes arrive in has no engines or tail section as it did in Planet of the Apes (1968) (it sank under water) or Brent's in Beneath the Planet of the Apes (1970) (it was burnt up). It therefore cannot possibly be either ship.
The torn spot on the bag of oranges changes position between camera shots.
When Dr. Hasslein opens the trunk of his car to retrieve the binoculars, he is seen removing the keys to the trunk from his pocket (this, in itself, is an error as he just parked; we're to believe he put the keys in his pocket whilst still seated in the car). In a subsequent shot where he opens the trunk again (to retrieve his revolver), he retrieves no keys from his pocket and the trunk key is already in the trunk lock and clearly visible, having been there when he parked.
In the shot immediately before Zira faints in the museum, the top button of her cape is fastened incorrectly. In the following shot as she is being helped up, the button is correctly fastened.
We are supposed to believe Zira convinced the chimp in the circus to give up her baby and take her baby in exchange for keeping Zira's baby. Mother's in the animal species are very protective of their young. It is not believable this would have occurred.
(at around 51 mins) In a close-up during the interrogation scene, Kim Hunter's own mouth and teeth are visible beneath the make-up of Zira's face.
(at around 5 mins) During the opening titles of the movie, when the helicopter is "lifting off" from the beach, it is the same footage of the landing, played backwards.
The capsule that the Apes exit at the beginning of the film does not have enough mass/length, that conforms to the interior of the ship that was established in Planet of the Apes (1968).
As the Ape-onauts exit the beached capsule, the hatch opens on the left side (SCREEN right) of the craft which completely distorts the interior of the capsule that was established in Planet of the Apes (1968). By all accounts the escape hatch opens right in the middle of the space where Dodge & Landon's hibernation chamber were located which is impossible.
(at around 6 mins) When Cornelius, Zira and Milo are first taken to the zoo, Roddy McDowall's neck is visible underneath his makeup.
The ape world is an underdeveloped and primitive society that believe flight is not possible according to the first film. Yet, three apes were able to find Taylor's ship, raise it from the depths of a lake, dry it out completely right down to the electronic equipment, figure out how to fly it, then finally enter a time warp to bring themselves to 20th century Earth. (Note: Apparently, Dr. Milo - after raising the ship from the water (most likely with help) - studied the ship's technical manuals. Entering the time warp was accidental when the Alpha and Omega bomb had detonated while the ship had been in flight).
While they were celebrities, Cornelius and Zira were taken separately on tours. It had been known that Cornelius was an archaeologist, but it was inexplicably Zira who had been guided around the Museum of Natural History (she looked bored).
(at around 27 mins) Whilst in captivity, Cornelius explains that Dr. Milo salvaged Taylor's space ship (which crashed and then sank in Planet of the Apes (1968)) and rebuilt it, which allowed the three apes to escape from the Earth before it was destroyed and travel backwards in time. The apes in the 40th century did not have the technology to salvage the ship from the bottom of the lake, let alone the knowledge to rebuild it and learn how to fly it. It is even more preposterous that they would have been able to fly it *back* through the time warp with such precision. (Note: Cornelius' exact words were "It was Dr. Milo who salvaged it, studied it and half understood it". Once in orbit, the shock waves from the explosion of the Alpha and Omega bomb plunged the ship into the time warp, unexpectedly).
When they are feeding oranges to the chimps, the soldier mentions them getting dressed and having a suitcase. At this point, a boom mic shadow is visible as it passes from the general to the soldier, and is visible on his tan shirt.
At the beginning of the film, when the "astronauts" emerge from the capsule in full spacesuits and helmets, they are walking fully erect, unlike the hunched over gait of the apes in Planet of the Apes (1968) and Beneath the Planet of the Apes (1970).