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)
Visible crew/equipment: While Sarek is lying in sickbay, McCoy, Spock, and Kirk discuss the open heart procedure, and when Christine walks over to stand beside Sarek, the white tape mark is visible on the floor where she stands. (00:24:20)
Visible crew/equipment: While Spock and McCoy are discussing the dangers of the experimental drug, when McCoy tells Spock that he can't sanction it, the two white tape marks are visible on the floor. (00:26:30)
Continuity mistake: During the fight with the Andorian Kirk was stabbed in his lower kidney region, he even holds his hand there to stop the bleeding. In sickbay later on, he has a bandage around his chest and McCoy says he has a punctured lung. (00:28:30)
Continuity mistake: Just before Kirk kicks his attacker for the final time during the fight in the corridor, the Andorian picks the knife up off the deck with his left hand. In the very next shot, the knife has instantly switched to his right hand. (00:28:30)
Visible crew/equipment: When Kirk regains consciousness in sickbay he asks McCoy about Sarek, and as the camera turns we see the white tape mark on the floor in front of Amanda's feet. (00:35:20)
Continuity mistake: After their life threatening surgery, Spock and Sarek somehow manage to switch beds between being in surgery and recovering. (00:45:00)
Continuity mistake: In one scene the Enterprise's phasers were pink instead of blue.
Other mistake: 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.
Suggested correction: This was an intended physical effect to simulate the cryogenic surgery that McCoy was performing on Sarek.
Character mistake: When Kirk turns to face Spock and McCoy to introduce them to Sarek, the hangar deck nameplate on the wall is misspelled. It reads "hanger deck".
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.
Continuity mistake: In sick bay, close ups of Sarek show him sitting up at a 90 degree angle, but the long shots show him reclining at a 45 degree angle as the shots cut away and back.
Continuity mistake: In the reception room, Amanda's glass goes from 1/4 filled after McCoy poured more, to over 3/4 full without his adding more.