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.
Suggested correction: According to the captain's log at the beginning, they have been on Vulcan for 3 months. As they prepare to depart, we see several Vulcan technicians moving equipment in and around the ship. It's quite conceivable that the bridge was reconfigured according to the crew's specifications to facilitate their use of the ship. This may seem a bit excessive, as the remodel includes the door onto the bridge, and the frame of the door, and possibly most of the rest of the ship. But it's not outside the realm of possibility. The real mistake, though, is why they would go to the effort of installing new workstations on the Klingon bridge and marking them with Klingon labels, instead of standard Federation text. Interestingly, though, the "Starfleet" style bridge layout of the Klingon ship is being used by the Klingons on the Bird of Prey in Star Trek V.
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)
Continuity mistake: Notice the candle on the table in the Italian restaurant scene during Capt. Kirk and Dr. Gillian dialog. It changes height depending who is talking. (01:03:50 - 01:08:30)
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)
Continuity mistake: In the operating room, when Kirk uses a phaser to melt the lock, the hand he is holding the phaser in switches hands in the closeup shot. Additionally, the arm holding the weapon in the closeup is no longer in hospital scrubs, but a similarly colored sweatshirt. (01:25:20)
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)
Continuity mistake: In the scene where Spock gets confronted after swimming with the whales: you can easily see before he puts on a robe, that his clothes are completely dry, yet he had just been swimming in the shot a moment earlier.
Continuity mistake: The crew wrote 'HMS Bounty' in large red letters on the side of the Klingon Bird of Prey, but in all subsequent flying shots, the large red writing is never seen again. It's not burned off as you only get heat entering the atmosphere if the speed is sufficient, and it's not wind resistance either, because the lettering is missing just after they take off.
Continuity mistake: In the scene with Kirk and Spock walking the marina green under the Golden Gate Bridge, the leading shot shows blue sky. As the camera changes angles, the background changes to completely fogged in. Not shot on the same day.
Continuity mistake: When we first see the whale pen it is divided from the sea by a concert wall with double gate in the middle, but when the whales are released most of the wall and gates are missing.