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.
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 the beginning of the film (around the 11 minute mark) 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 16 minutes) 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.
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.
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.