Right before Caesar gets tortured by electrocution, the operator throws 3 of the 4 switches on the control panel. He then starts moving the dials. In the close up all 4 switches have suddenly been thrown.
When Armando is being questioned by Governor Breck and his men, Armando is shown the taped conclusion from the Presidential Commission regarding the fate of Cornelius & Zira (footage from Escape from the Planet of the Apes (1971)) on a round video monitor, the image changes from an approx. 1.33:1 aspect ratio, then to a wider aspect approx. 1.85:1 in the close-up of the monitor, then to a circular image in a following wide shot and then finally back to the approx. 1.85:1 aspect in the return close-up of the monitor. (Note: the differing aspects are the result of some portions of John Randolph's performance consisting of stock footage from the previous film combined with new footage filmed for this movie).
(at around 1h 16 mins) When Caesar gets an M-16 during the riot from the armory, he's running along firing. As he's shooting, two riot police with shotguns are firing back at Caesar. The officer on the right of the screen has his shield down, but in the very next shot as he's being killed, his shield is raised.
Early on, when people are protesting in front of a government building, marching in a circle and holding signs, there is an announcement on the loud-speaker being made. The announcement continues without interruption, however, the people marching in the circle cut from two people to two different people in the same position in the circle without disruption of announcement audio.
In less than 20 years since Escape from the Planet of the Apes (1971), chimpanzees, gorillas and orangutans have evolved to a perfectly humanoid form, a progression that would take thousands of years, not just one generation.
It's mentioned that some apes are being shipped to "French Cameroon." That part of Cameroon became independent of France on January 1, 1960, with the rest becoming independent of the United Kingdom on October 1, 1961, unifying into the Republic of Cameroon that exists today. Meaning, Cameroon had been completely independent for over ten years by the time the film was released.
Armando calls the son of Zira and Cornelius "Caesar" at the start of the film rather than his given name, "Milo". Since this name was well known to the authorities as that of one of the three chimps that had arrived from space in Escape from the Planet of the Apes (1971), this was a wise move. When Breck was having Caesar choose a name for himself, Caesar went through the book of names clumsily, as an illiterate ape would, while quickly scanning for his own given name, making it appear that he chose the name "Caesar" randomly.
(at around 54 mins) When Caesar is being tortured, a small rectangular hole in his mask at the rear of the upper lip is visible, presumably for Roddy McDowall to breathe through.
(at around 48 mins) Throughout the film, the supposedly wireless telephones are simply standard telephones with the wires removed and the wire outlets taped over. This is most noticeable when Mr. McDonald is talking on a red phone and the tape color does not match. This is due to some passages of John Randolph's speech being stock footage and some having been shot for this film.
Throughout the film, the apes have human-looking feet, and not the ape-shaped feet as in the earlier films. This is most apparent when the apes are strapped to the electrical torture device, to which the feet are visibly bound.
Near the end of the film, and at night, when the apes are beginning to rise up, the city center is shown with first, people running to take cover, and then military forces taking positions, while the apes are shown spilling gas around the area, almost none of the very visible public lighting is shown lit.
Apes, especially gorillas, are shown to struggle in physical combat against humans, while actual gorillas are very strong and heavy, and could easily tear a human apart.
At 1:09:51 time stamp remaining mark, the Presidential Commission spokesperson says that Zira was pregnant with child as opposed to saying a chimpanzee.
(at around 1h 11 mins) When the apes rise up against the humans, shortly after an ape throws a flaming torch to start a fire, the camera pans back and for a brief moment you can see a metal tube mounted to the ground shooting gas to create the wall of fire.
Most of the film is taken up with the hunt for Caesar, presumably so he can be killed to prevent the future revealed by his parents. However, at no time is it considered that he may already have fathered numerous children who would continue his bloodline.
In the Command Post, a human worker hands Caesar a red paper pack and says "DO". Even the most intelligent human would not have the slightest idea what to "do" when handed a random item and expected to "know" what to do with it. (Note: Undoubtedly, the Command Post slaves had received training off-screen as to their work function).
DNA testing, which was in its infancy at the time of the filming, was apparently unknown to the writers of the film. If it wasn't, then discovering who Caesar was would have been a relatively simple task .