Continuity mistake: After the bird of prey first fires on Enterprise, Kirk is thrown right out of his chair to his left and lands on the floor. The camera pans around the bridge briefly, seeing the rest of the crew staggering and holding onto things. In the very next shot Kirk is back in his chair shouting "back off, back off" and the crew back in their positions as if nothing has happened.
Deliberate mistake: It's not particularly believable that the away team to stop the assassination would consist of the ship's entire senior staff, and raises the question of who was left in charge of the bridge (the answer: apparently nobody, as they return to a completely deserted bridge). Of course this is dramatic license, and at this point the audience won't care or likely even notice.
Other mistake: Throughout the film, Valeris is referred to as a Starfleet Lieutenant. However, in every scene, she wears the rank of Commander on her uniform. Best comparison can be made in Star Trek II: The Wrath of Khan where the same uniforms were used. Saavik (Kirstie Alley) wore a Lieutenant rank (2 piece), where Chekov and Scotty (himself then still a Commander, until promoted by Admiral Morrow in Star Trek III) wore the single-piece rank insignia used by Valeris in The Undiscovered Country.
Continuity mistake: It's slight, but noticeable. The nails used to keep Chang's eye patch in place change slightly over the course of the film. For example, when they're beamed aboard the Enterprise, the nail closest to Chang's eye is at the top of the patch. But, during the climactic battle, it's down, closer to his nose.
Other mistake: It was shown that whatever form Martia was in, whether it be the attractive semi-human form, the Sasquatch form, or the little girl form, her voice always remained the same. Why didn't her voice remain when she morphed into Kirk?
Suggested correction: Clearly she has the ability to control when and how her voice changes in the same way she controls her physical form.
Character mistake: There's no apparent reason for Spock's conviction that the gravity boots are a damning piece of evidence that will reveal the assassins, especially considering how useless they prove to be after being found planted in an innocent man's locker.
Character mistake: At the trial, the witness said that "With the first shot, we lost our gravitational field. I found myself weightless, and unable to function." However, the first shot caused the ship to start listing to port. It was the second shot that caused the gravitational failure.
Suggested correction: People misremember events all the time, especially if the events are unexpected and chaotic. At best a character mistake.
I called it a character mistake.
He could also simply be lying in order to explain why he did not do whatever he should have after the 1st shot, since he was an easy target for the intruders and did not die in battle, which would have been honorable. Klingons are not the most honest of people.
Continuity mistake: Valeris seems to appear out of nowhere during the thwarted assassination attempt on Khitomer. The Enterprise crew beam down without her, spring into action, and when Admiral Cartwright yells "Arrest those men!" Spock emerges from the crowd with Valeris in his custody as if she were always there with him or he'd pulled her out of a hat.
Character mistake: Chekov pulls a gun on the Klingon ambassador even though he hasn't been implicated in the conspiracy.