Continuity mistake: Uhura's image on the conference viewer is of a close up on her face. When the camera looks at the screen, the image is of her head and upper torso. Likewise, when Fitzpatrick is on, first it's his head and torso, then it's only his head and shoulders.
Other mistake: The red alert light at the entrance to the chapel blinks a few times and stops, even though the rest are still flashing. Also, the last thing Kirk says is "All decks, condition red", which is unnecessary, since the red alert has already begun sounding.
Revealing mistake: After the ship enters the galactic barrier, when Kirk orders "gravitation on automatic" there are flashes of light, and in a closeup of the first console just to the left of the main viewscreen, we can see the special effects material spread across the panel below the controls, for the fire to burn when it explodes. Then it cuts to the wideshot as a second console (near Spock's station) blows up, but in the background that first console is intact - no fire/damage whatsoever (it explodes again in a following wideshot).
Continuity mistake: When Kirk exits the turbolift he sits in the captain's chair, and Yeoman Smith stands behind him with barely any space between them, but in the shot facing the main viewscreen Smith is gone, and in the next shot she reappears right behind Kirk again.
Continuity mistake: During Mudd's hearing Eve's sitting in the chair beside Mudd, but when Kirk states that the hearing is closed and that Mud will be handed over to legal authorities, in the wideshot Eve's chair is empty, but in the next closeup Eve is still sitting in the chair.
Visible crew/equipment: When Spock walks into the briefing room, he stands in front of the closed doors telling himself that he's in control of his emotions, and when he says, "My duty," the shadow of the boom mic can be seen moving across the wall at the top of the screen.
Visible crew/equipment: After Kirk's briefing regarding Mitchell's abilities, when Spock and Kirk are alone just as Spock says that they'll never reach an earth base with Mitchell aboard, the boom mic casts a shadow overhead until Spock steps forward, and a bit can be seen again when Spock tells Kirk that it's his only choice.