Continuity mistake: During the attack on the outpost, as Spock arrives in his foxhole, Kirk puts down the box of ammunition and positions the grenade launcher. We cut to a higher angle as Spock scrambles around to the Captain's left, and Kirk is putting the box down and positioning the launcher all over again. As the action was supposedly continuous, there wasn't time for Kirk to pick the items up and set them down a second time. (00:08:40)
Continuity mistake: The lawgivers' robes that Kirk and Spock take off and drop on the floor just inside the doorway disappear after they phaser the wall to reveal Landru. The robes reappear a few minutes later, but have now migrated several feet to one side, bunched up against the wall where they are no longer in tripping range of the actors about to enter the scene. (00:42:25 - 00:45:00)
Continuity mistake: Kirk is holding three large tribbles while talking to Uhura on the bridge. After McCoy enters, the tribbles keep changing positions in Kirk's hands as the shots cut back and forth. (One on top and two underneath, then two on top, etc.). (00:31:45)
Continuity mistake: After partially healing him, Gem falls away from McCoy and lands, unconscious, perpendicular to the dais he's lying on. When Kirk and Spock rush over, however, she's suddenly moved several feet away and is now lying parallel to the dais. (00:44:50 - 00:45:35)
Continuity mistake: While Kirk and McCoy attack the other two henchmen, Spock shoulder-pinches the card-playing gangster. In the very next shot, Spock's position has jumped from behind the now-unconscious man to several feet away, where he is bending over to retrieve the guns from the floor. (00:16:30)
Continuity mistake: McCoy uses a "masking device" to block out the bridge personnel's heartbeats. When McCoy masks Spock's heartbeat, he holds the microphone, err, masking device over the wrong part of Spock's anatomy. We'd already established that Spock's heart wasn't in the same place as a human.
Continuity mistake: Natira orders the landing party to kneel before the oracle. Spock does so about a foot-and-a-half to McCoy's left. As the shots change, Spock "jumps" to McCoy's immediate side, then back to his original position, etc. (00:10:35)