Other mistake: After they take off from California, Kirk gives a heading to Alaska. He then tells Sulu "full impulse power", and Sulu says "aye, ETA 12 minutes." Full impulse is 1/4 the speed of light. No way they would use speeds like that to go a few thousand miles.
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.
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.
Factual error: When the Bounty is preparing to slingshot around the sun, the sun is burning a light yellow. From space, it would be white; it only appears yellow when viewed from Earth because of how the visible color spectrum is filtered through the atmosphere.
Factual error: The nuke ETs are sitting at a RADAR station, not a power output monitoring station. The RADAR wouldn't give any sort of indication of sudden additional power draw on the reactor. (01:12:00)
Revealing mistake: When the ship crash lands in the water, and they open the hatch you will see a wall, looks like some piping. When they show the outside of the ship no such wall exists.
