Factual error: Kirk, McCoy, and Spock are chained to the dungeon wall, with McCoy noticing he is next to a skeleton, also chained. This is incorrect, as a skeleton's bones are held together by cartilage, which would rot over time. Therefore, the skeleton would not be hanging intact, but would be a pile of bones on the floor.
Suggested correction: As we learn in the episode, the 2 aliens created the entire castle only recently to appeal to the human's primal fears and superstitions. A pile of bones on the floor would not have the frightening impact of a hanging skeleton.
Visible crew/equipment: A stagehand's foot is visible before the party beams down. (00:09:10)
Continuity mistake: Kirk and the others are beamed down to the planet, and when the Enterprise crew are led into Mudd's throne room by two Alice androids (the Alice series are 500 identical androids) they are wearing the usual metallic, backless, kitten heel slip-ons, but in the next shot android Alice at the left is now wearing altogether different shoes - cream color with ankle-straps. (00:09:30)
Other mistake: When Norman goes to the second panel in Auxiliary Control, there is clicking when he turns his right hand, but there is no knob, just a blinking light.
Plot hole: How did the android Norman get aboard the enterprise? If he beamed aboard I'm sure someone would have noticed and where did he beam from? The Enterprise was nowhere near any planet and I'm sure they would have detected any spacecraft nearby.
Suggested correction: These are unanswered questions, not plot holes.
But something phrased as a question because it has no decent answer can constitute a plot hole.
Continuity mistake: In Norman's chamber, the medallions on Alice 3 and 11 go from blinking to steady, to totally off and back to blinking through the scene.
Continuity mistake: When Spock shows the others the machine which must be put "in the proximity of the Companion", in the closeup Spock's left arm is up on the machine and we see the chair opposite Spock is empty. However, in the next shot Spock's right arm is up by the machine and Kirk is sitting in that chair. (00:22:10)
Continuity mistake: The shiny gold blanket covering the dying Commissioner Hedford is tucked up to her neck in close-ups, but only reaches her upper waist in full shots. (00:36:45)
Continuity mistake: Kirk is holding the communication device out to his right from his view of the Companion and Cochran's, but the next instant shot shows Kirk holding the device close to his chest as he speaks with the Companion.
Continuity mistake: The designs of the fabric the Companion holds up to view Cochran are different from her view versus the view from Cochran's.
Continuity mistake: After landing on the planet, the interior walls of the shuttle craft or well-lit, and light green in color. As Kirk and party exit, the interior is seen to be dark and gray.
Visible crew/equipment: At the hangar deck, after Kirk introduces Sarek to Spock and McCoy they all step forward, then as Sarek introduces his aides and his wife, the three white tape marks are visible on the floor behind them, where they had been standing. (00:02:15)
Visible crew/equipment: After Amanda tells McCoy about Spock's pet sehlat, when Sarek and Amanda walk away, the actors' multiple white tape marks are visible on the floor, under the table. (00:10:50)
Continuity mistake: When Sarek enters the banquet room to take his medication, the Tellarite sitting at the table holds a glass in his right hand. In all the full shots with Sarek in the foreground, the glass switches to the Tellarite's left hand. In all the close-ups of him, it's back in his right again. (00:14:50)
