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: After Kirk and the landing party beam down, they are met by a man who tells them where they can stay for the night but they will have to hurry because it's almost the "red hour." At that time the camera shows a clock that reads 5:58. After being introduced to the daughter of the man who can give them shelter, a gong rings and magically it is 6:00, 2 minutes later, although only 32 seconds of real time has passed. (00:05:18 - 00:05:50)
Visible crew/equipment: After Scotty tells Kirk that the shields are unable to deflect the heat beams, the piercing soundwaves knock everyone out, and when Kirk awakens in the cell he tries to wake Spock, then the shadow of the camera is visible on the pillar as it follows Kirk. (00:25:55)
Other mistake: On the bridge, as the camera looks at the view screen, there is a flashing blue light reflecting on the console in front of Sulu, but there is no light other than scanner under the viewer flashing and it's too far away to project light that far, and it's amber colored.
Continuity mistake: While Kirk is attempting to convince Marplon and Reger to take them to Landru, a distant shot shows Marplon's right hand descend from his chest to his side. Immediately thereafter, a close-up shows Marplon's hand again suddenly at his chest.
Other mistake: After Sulu is "absorbed", he is beamed up to the ship. After materializing, his demeanor could be described as calm, gentle, meek and euphoric. After that scene, the captain's log quotes Kirk as saying "Mr. Sulu has returned, but in a highly agitated mental state." Was Kirk looking at a different Sulu than we were? A gentle, meek and euphoric condition incorrectly being touted as "highly agitated"? Time for Shore Leave. (00:01:28 - 00:03:53)