(at around 16 mins) As Brent and Nova start to ride away from the wrecked spaceship, a white paper cup blows from the camera position. It blows along the ground into the center of the frame and then around behind the horse.
(at around 57 mins) When Brent first encounters a mutant, Verger, he can see the "alpha omega fins" of the doomsday bomb. He turns and runs back to the double doors (2 seconds), turns around to respond to Verger's "beeps", and in this impossibly short time the bomb has already been lowered into the silo, so only the top is visible.
(at around 38 mins) When Brent tries to get the reins from the runaway horses you can see an extra set of reins going into the paddy wagon through the holes in the wall, to a hidden driver. After Brent stops and dismounts, the reins disappear.
(at around 11 mins) At the beginning of the film, where Brent meets Nova for the first time, 2 taped position markings (bright white letter T's) can clearly be seen on the ground behind Brent. These markers appear and disappear between shots.
(at around 22 mins) When Brent is shot in the arm by a gorilla patrolman, a bird is seen flying away. Yet in the first Planet of the Apes movie, Cornelius and Zira argue with Taylor that flight is a scientific impossibility and Taylor makes a paper airplane to prove his point.
Brent discovers Taylor's connection to Nova by her having his dog tags. In the first film he does not have dog tags when he is captured by the apes, or when he rides off at the end of the film. There is nowhere he would have gotten them since his ship sank and his supplies and clothes were destroyed by the indigenous humans.
In Planet of the Apes, Zira and Cornelius stated they were planning to be married, but in this film they are already married. This film begins right where the old one ended, so there wasn't time in between movies for them to have been married.
Besides shock value and a twist, there's no logic as to why the mutant humans would wear "normal" human masks. They have looked like this for hundreds/thousands of years. There's no one they need to fool. Since it is just the apes, themselves, and primitive humans.
(at around 20 mins) During Ursus' speech to the Ape Council, James Gregory is very animated. He is so animated, in fact, that near the end of the speech he turns and lifts his head, revealing his throat and Adam's apple.
(at around 1h 30 mins) When the soldiers bring down the A-Bomb, they throw ropes over and behind the casing. They then pull it down. The bomb is eased down gently, yet the fact that the rope wasn't looped around, and it was not slowed down by gorillas lowering it from behind would have led the bomb to just topple to the ground, revealing that it was either filmed as it was lowered down by hidden wires, or it was on a tilting rig that was fastened to the ground.