Continuity mistake: During the attack on the outpost, as Spock arrives in his foxhole, Kirk puts down the box of ammunition and positions the grenade launcher. We cut to a higher angle as Spock scrambles around to the Captain's left, and Kirk is putting the box down and positioning the launcher all over again. As the action was supposedly continuous, there wasn't time for Kirk to pick the items up and set them down a second time.
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.
Revealing mistake: When Kirk finds his position on the bridge occupied by the Tribbles, he calls for Dr. McCoy, and as he arrives, Kirk thrusts some of the Tribbles under his nose. In the shots where Kirk's face is seen, one of the Tribbles he is holding features a visible seam.
Continuity mistake: When the landing party beams over at the beginning, the transporting officer is wearing a blue jumpsuit, but when they go to the closeup of his hands, they have a red uniform, two braid stripes, and are the standard shot they use when Scotty does the transporting.
Continuity mistake: Spock states that the Fesarius (Balok's ship) "must be a mile in diameter". Yet any one of the small spheres that make up the Fesarius dwarfs the Enterprise. If the Enterprise is about 300m long, the Fesarius would have to be around 6 km in diameter - considerably larger than a mile.
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: The bundle of clothes Kirk steals from the balcony changes several times while he and Spock face the policeman. The red shirt is not visible, then is hanging down on the right, then isn't. It then disappears from view until they reach Edith's basement, where Kirk puts it on. Two belts, two jackets and Spock's cap also seem to have added themselves to the bundle somehow. They're not there when Kirk first dismantles the tangle of clothes in the basement.
Plot hole: Apparently there is some confusion over the distance between Earth, Starbase 11, and Talos IV. When Spock first meets Pike on Starbase 11 he tells Pike Talos IV is only six days away. Yet when Pike (in the recording) speaks to the Talosians for the first time, he says he is from a star system on the other side of the galaxy. If Talos IV was on one side of the galaxy and Earth was on the other side, it would take hundreds of years at maximum warp to travel from one planet to the other.
Character mistake: Spock tells Kirk that without being able to test their vaccine with the ship's computers, "it could be a beaker of death." The vessel that the vaccine is contained in is a flask, not a beaker. The ship's science officer should know the difference.