Continuity mistake: During McCoy's struggle with the orderly, only half of the plastic colored bottles strewn on the table are knocked off onto the floor. But when Nurse Chapel hurries over to tranquilize the orderly, the rest of the bottles vanish, leaving the table completely clear. (00:19:10)
Other mistake: After Chekov goes beserk on the bridge and Dr McCoy and Scotty restrain him, Spock enters the scene and grabs Chekov by the sides of his face. His hands are close together as a human's face is around eight inches wide. When the camera cuts to Chekov's perspective view of Spock, Spock's arms are spread way far apart. Either Chekov's face has ballooned to over two feet wide or Spock is grasping the Panavision camera by its two sides. (00:14:34)
Other mistake: After the shot of McCoy checking Sickbay, there's a shot of Chekov. It was a quick shot but it can be seen that the film was run back and forth through about a second, as his movements can be seen to repeat rapidly in reverse and forward twice. (00:06:56)
Continuity mistake: When Kirk, Bones, Chekov, and Spock go over to the Defiant, the suits they have on have a black support bar in the left side of the headpiece. When the camera looks down at the dead men and back to Kirk, you see the bar is on the right, and his hair is parted on the right. The very next look at Kirk, the bar and the part of his hair is back on the left.