Continuity mistake: In the very first scene after the intro, when Bela's goons hold the landing party at gunpoint, Spock's position on the sidewalk shifts back and forth from standing on the other side of a wooden bench to standing alongside Kirk and McCoy with no bench in between. In fact, in the latter shots, the bench has moved all the way across the street.
Continuity mistake: When Kirk sits down to consult the computer, the big green potted plant on the shelf behind him is there in full shots and missing in close-ups. It's much too large to be completely hidden behind him: if it shows up behind his head in full shot, it should show in close-ups, too, but it doesn't. It's not there.
Continuity mistake: Mad Lazarus has just put a black-and-white hatch cover over the stolen dilithium crystal in his time ship. It's in place when Kirk arrives to confront him. But a moment later, when Kirk leans in and is accidentally transported to the negative universe, the cover has disappeared. It's still missing when Kirk returns to push Lazarus through the portal.
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: 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.
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.