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)
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)
Visible crew/equipment: The action is shifting back and forth from the bridge where the alien ship is attacking and sickbay where McCoy is operating on Sarek. In two separate takes when the scene shifts to sickbay McCoy is standing at the operating table. I noticed a wisp of smoke rising from McCoy's right side (left side of screen). It appears to be from a cigarette in an ash tray below camera level either beside McCoy or near the head of Spock's bed.
Continuity mistake: An Orion spy (disguised as an Andorian) attacks Captain Kirk in the corridor on Deck 5 and plainly stabs Kirk in the back on the right side. As he staggers to his feet, Kirk first reaches to the wound with his right hand, indicating that it is, indeed, on the right side of his back. He then reaches far across his back with his left hand to touch the wound, and we see blood on the back of his hand (but not on his sleeve). So, it is visually established that the wound is on the right side of his back. But, a few moments later in SickBay, Dr. McCoy announces that Kirk's wound is a "puncture to the left lung." Then, later in the episode, Kirk his holding his left arm as though he is suffering pain on the left side.