We first see Captain Leo Davidson training a chimpanzee in astronautics. Its inconceivable that an officer with such responsibility would be unaware of the taxonomic and cladistic difference between apes (including chimpanzees and humans) and monkeys (the New World and Old World monkeys). Yet he twice refers to his student as a "monkey".
(at around 1h 6 mins) As Leo begins to enter the water with Ari on his back, she puts her hands over his face. In the next shot from further across the water, they are by his side. The hands return over his face once they are in the water.
There are numerous apparent plot holes in the story which are explained away in supplementary material (DVD booklet, novelization, etc). This includes the frequently asked question regarding the presence of horses. These remain goofs as far as the casual viewer is concerned, but they are the subject of much heated debate and are not now included in this list.
How did humans from far and wide hear about "this human who defies the apes"? Leo was hardly noticed by anyone but the group he was with, and a few apes. The apes wouldn't publicize that a human was making fools of them, so how did word of his presence spread??
(at around 45 mins) As Leo and co. escape the city through the tunnels, they break through some boards covering the tunnel exit. One of the boards has been clearly sawn through and is broken at that point.
(at around 1h 40 mins) When Leo is about to trap Thade inside the wreck of the Oberon, the touch screen that activates the shield is shown twice. It is clearly encrusted with a thousand years of dust. But as Leo actually puts his hand over the screen it is suddenly as clean as the day it was made.
Many people believe that, when the group goes to the crashed spacecraft, there are rock formations in the aircraft. This would not be possible, since it takes millions of years and a lot of pressure to form rocks. The spacecraft, however could have landed on rocks that were there before the spacecraft crashed.