Revealing mistake: When Kirk enters Spock's quarters at the beginning of the movie, he hears Spock's voice calling out to him from a darkened area. As he walks into Spock's quarters, pay attention to the top right area of the screen. You'll notice the top of the walls and no ceiling. That lets you know that they were filming on a sound stage in a set with no ceiling. (00:14:50)
Revealing mistake: When the Enterprise explodes, some of the debris briefly turns transparent. (01:12:15)
Revealing mistake: As the Enterprise nears the space dock doors there is an outline around the ends of the nacelles that precedes them as it moves. This shows that the ship was superimposed over the shot of the inside of the space dock.
Revealing mistake: On the genesis planet, after Kirk kicks Kruge into the lava, he approaches Spock lying on the ground and kneels beside him. It is obvious that William Shatner is wearing knee pads under his uniform trousers. In fact, they're visible for most of Kirk's time on the planet. (01:24:13)