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)
Continuity mistake: When Kirk checks the video logs to find the keeper of Spock's katra, the timestamp reveals that Spock melded with McCoy on stardate 8128.78. The Wrath of Khan begins on stardate 8130.3. (00:21:50)
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)
Suggested correction: I reckon the writers always refer to levels of "impulse power" precisely so they don't have to worry too much about particular speeds (personally I always thought of it as roughly analogous to gears on a vehicle, but your mileage may vary). They use impulse to leave dock in both Star Trek II: The Wrath of Khan and Star Trek VI: The Undiscovered Country (it's implied to be unusual in both cases, for what it's worth). If all of that contradicts an earlier episode, I think we're looking at more of a retcon situation than a mistake.
Suggested correction: The warp scale has been adjusted several times, so it is impossible to say precisely how fast this fictional technology is, and by extension, how fast impulse is.
Impulse drive speed on starships have been consistent. Although sometimes quarter impulse on a shuttle refers to quarter power and not speed. Even if the speed of quarter impulse is 10 times slower than suggested (and used in the series), spacedock would still be 1.3-1.5 times bigger than Earth, which it wasn't. "It's fictional technology" is usually only a valid correction if the technology isn't explained in-universe. However, when certain parameters regarding fictional technology are established (even if they set wide parameters such as warp speed velocities) violations or contradictions (through bad script writing or whatnot) are valid mistakes.
Continuity mistake: In the scene early in the movie where Kirk is talking to Sarek the guns on the wall are suddenly rearranged. (00:23:50 - 00:24:45)
Continuity mistake: When David and Saavik approach Spock's tube on Genesis, you can see quite a bit of dirt covering the words' red lettering, but when it does another shot of the lettering, it is noticeably cleaner. (00:30:25)
Factual error: When the sun sets on Genesis, it happens too quickly for a planet with an atmosphere, as well as having a crisp shadow that it wouldn't have. The view of the sun as it's setting is that of a planet with no atmosphere. (00:54:00)
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: When Kirk and Kruge are fighting, a piece of the cliff breaks away and wiggles down the side of the cliff rather than falling like a rock. (01:22:55)
Revealing mistake: Obvious stunt double for Kruge during the fight with Kirk. (01:23:00)
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.
Suggested correction: This isn't wrong, per se, but matte outlines are so inherent to this type of optical effect that it I think it would be like counting "makeup looks fake" or "unconvincing acting" as mistakes. There probably is a line where a special effect is so bad it can indeed be counted as a mistake, but this doesn't cross it, in my opinion.
Suggested correction: The stardate system has never been precisely defined, so this is not a mistake.
While not precisely defined, it has been established in almost every episode that the numbers increase as time moves forward. (example in the Next Generation, the second number of the stardate corresponded with the season number, also in the episode The Best of Both Worlds, Picard gives a Stardate of 43992.6, then later 43996.2. So this mistake stands.