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)
Continuity mistake: The scientists tell Zira to put the shaped blocks back in the box. She starts by placing an orange heptagon in the box, followed by a purple square and blue square, followed by a blue triangle. This is all she puts in the box in this shot, but by the next shot, a red star has appeared and an orange shape has appeared in the corner of the box. Also, the blue square has vanished and Zira then puts a purple box down in its place. (00:14:10)
Factual error: Zira puts the blocks together to form the stairs that will allow her to get the banana, but only stares at it. When the female scientist asks why she won't eat it, Zira says she loathes bananas. Bananas grow in tropical climates near the equator. New York City, where she lived in the future, is in a subtropical climate zone, and not grown there. How would she loathe bananas if she's never eaten one before?