Factual error: According to "The Doomsday Machine", full impulse drive is one-quarter the speed of light. In the first two movies, Enterprise used thrusters as opposed to impulse drive to leave Spacedock, confirming the notion that impulse drive is far too fast to leave such a (comparatively) small structure. Styles, however, orders Excelsior to one-quarter impulse, which is 18,750 km/s. In one second, she will travel half again Earth's diameter. From the time he gives the order to the time we see Excelsior clear spacedock's doors is approximately 40 seconds. Even allowing 30 seconds to go from rest to one quarter impulse, spacedock must be 13-15 times bigger than Earth! That's some serious engineering.
Continuity mistake: When Kirk, McCoy and Sulu beam aboard the Enterprise we switch to an outside shot and the bridge lights come on, a very bright white light that lights up all three windows, but when we go inside the bridge is very dark with only a few lights on.The main lights come on when they man their stations.
Continuity mistake: When Savvik and David first encounter Spock's photon tube "casket", they open it to reveal the burial robe rather nicely folded with the Vulcan lettering shown. When David reaches in to retrieve it, the robe is very disheveled, lying more lengthwise, and the lettering not so clear.
Continuity mistake: Early in the film when a lifeform was detected in Mr Spock's quarters, you see a blip on the screen inside the old Enterprise, not the new Enterprise, which was almost a complete rebuild, with a very different layout. Notice the outline of the ship - different engines.
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.
Other mistake: On the backside of the DVD near the bottom is a series of six images purportedly from the movie. The second image shows the crew in uniform, including Mr. Spock, standing in the Federation Council chamber. This image comes from the movie Star Trek IV: The Voyage Home (after the crew has returned to Earth and is waiting to hear the Council's judgment).