Continuity mistake: The Enterprise looks way more damaged than it did at the end of II. In the Wrath of Khan, Enterprise takes nearly all her damage on her port (left) side. At the start of The Search for Spock, she is damaged about equally on both sides, and still more so. (00:12:05 - 00:13:20)
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. (00:13:30)
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)
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. (00:23:45)
Other mistake: When Savic is sleeping, a burst of energy from planet pushes the tree she is leaning on upward. There is a Klingon above that almost gets hit with it. Savic should have seen him and started moving with Spock. This might have been a scene meant to be used later at the climax. (00:57:00)
Revealing mistake: When the Enterprise explodes, some of the debris briefly turns transparent. (01:12:15)
Visible crew/equipment: When the Enterprise is exploding, a Klingon is shown flying over the helm. When he lands, a person's arm can be seen on the right side of the screen, trying to grab his foot. (01:14:50)
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)
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.
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.
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).