Deliberate mistake: Nothing in space could communicate with whales in the ocean without radios. It wouldn't matter if it was generating the loudest sound in the universe, or had the most sensitive mike and most powerful amplifier - sound can't travel through a vacuum. The probe couldn't "hear" the whales, and the whales couldn't hear the probe.
Continuity mistake: The Bird of Prey is the one captured by Kirk's crew in ST III That ship's bridge showed Klingon Cmdr Kruge in his elevated command chair with his helmsmen arrayed circularly below him, and nothing else. ST IV has this same ship, however the bridge now resembles The Enterprise layout with Kirk's command chair behind Sulu and Chekov at their rectangular helm, with Spock, Scotty and Uhura at their usual positions.
Visible crew/equipment: In the scene where Sulu flies the helicopter and accidentally turns on the windshield wiper, look closely at the bottom of the wiper and you can see the crewman's finger manually moving the wiper. (01:18:55)
Continuity mistake: Gillian's Chevrolet pick-up truck is obviously two different trucks, depending on its location. When near the aquarium, the grille is a series of large, rectangular openings. But when at the park, where the Bird of Prey landed, the grille is a much tighter series of more numerous, smaller rectangles, made up of much thinner bars. This difference is consistent through the entire film. (00:55:05)
Other mistake: During the inquiry at the start of the movie, the Klingon Ambassador is going over the footage of the destruction of the Enterprise from all the exterior views. This is very nice, but how would they have access to all of these different viewpoints of the destruction? Bearing in mind no-one was there to film it. It couldn't have been the Klingon Bird of Prey that filmed it, because in most of the destruction scenes, the Bird of Prey would have been no-where near or in a position to film it, and was of course captured and still being commanded by Kirk during the course of the inquiry. (00:04:10)
Revealing mistake: When Chekhov falls on the aircraft carrier a cloud of dust flies up as if he's landed on a mat or soft cargo. When we see him on the floor it's concrete.
Visible crew/equipment: In the scene where Mr. Chekhov is being interrogated on the USS Enterprise (the aircraft carrier) after being caught on board, a studio microphone being used for the shot is visible hanging above him. (01:14:45)
Continuity mistake: When the 'Bounty' materializes right above the whaling ship, it casts it into shadow. But the close-ups of the captain and navigator as they frantically turn the ship around reveal no trace of a deeper shadow on their faces and the surrounding waters; the lighting remains the same as in the scenes when they try to catch the whales. (01:33:45)
Factual error: When Chekov and Uhura first see USS Enterprise and then later beam aboard, the carrier shown is not USS Enterprise (a nuclear aircraft carrier), but rather USS Ranger (a conventional or oil-fired carrier). The most obvious clue is the "island", the huge structure that sits on the deck of the carrier. USS Enterprise has the most distinctive and unique island of all the USN carriers, being totally cube-like in appearance. The carrier in Star Trek IV has a more conventional style island. [It is true that the USS Ranger was used in lieu of the USS Enterprise for the movie. This is because the Enterprise was on deployment at the time. Though by the time the movie was made the Enterprise had had its distinctive pagoda-like island rebuilt by Puget Sound Naval Shipyard in 1980, the aircraft carrier served until 2012 with a still-distinctive cube shaped island.] (00:53:30)
Factual error: The "whaling boat" is too small to function as such. It isn't large enough to hold a fin, let alone disassemble a humpback whale.
Other mistake: When we first see the Klingon bird of prey on Vulcan (just before we see closeups of the "Bounty" written on the side) it is clear that the ship is far too small to contain a pair of adult humpback whales plus 18000 cubic feet of water in its hold. Later, when the ship is hovering above the sea after foiling the whaling ship's attack it seems considerably larger.
Continuity mistake: In the scene where Kirk and Spock are walking down the street after Spock has been in the tank with the whales it is clear that it is a windy day. Both of them have their hair blowing in the wind and yet as they are offered a ride in the truck and the shot leaves them for a moment their hair is suddenly fixed again. (00:55:20)
Factual error: They are about to perform major brain surgery on Chekov, but he is not nearly prepped for such a procedure. His head is not shaved, the area is not marked or masked and it has not been swabbed with Betadine, even though they are about to cut into his skull with a bonesaw. The "they haven't had time to prepare yet" argument doesn't hold water, because the surgeon is already approaching Chekov's head with the saw. (01:24:30)
Factual error: At two points in the film, the Klingon vessel Bounty traverses the distance from the Earth to the Sun at maximum speed, in excess of Warp 9. The latter sequence requires about 111 seconds from the time Sulu says "Aye sir, warp speed" until they reach the Sun. While there is no real science behind Star Trek's "warp technology, " the Starfleet Technical Manual provides a formula for calculating warp speed, whereby Warp 9 translates to about 136 MILLION MILES PER SECOND. The Earth is only 93 million miles from the Sun. In the apparent time that it takes the Bounty to reach the Sun, a starship traveling at such ferocious velocity would already be outside of our known solar system and deep into interstellar space.