Character mistake: When Kor describes the effects of the mind-sifter to Kirk (who is still in disguise as an Organian), he says the machine would leave him "more vegetable than human." There is no reason a Klingon would reference humans while speaking to a person he believed to be Organian.
A Piece of the Action - S2-E17
Revealing mistake: The drawers of Krako's desk are on the front side, not on Krako's side where they should be.
The City on the Edge of Forever - S1-E29
Factual error: Spock picks a lock during the great depression, but the lock is from the late 50's.
Continuity mistake: The second time McCoy is shown in the Sickbay, he is first shown sitting and writing. A moment later, he's standing bent over the terminal.
The Return of the Archons - S1-E22
Audio problem: In the transporter room, as Kirk is talking to Sulu, the door opens, McCoy comes in, and the door shuts. Trouble is that the characteristic sound of the door is missing.
Continuity mistake: In the finale of the fight between Kirk and Finney, Kirk's shirt is torn, exposing his right shoulder and arm. As Kirk (the stunt double) goes for the knock out blow, the shirt is only a little torn in the front. Then the close up on Kirk catching Finney, his shoulder is exposed again.
Continuity mistake: In the dining hall, Korab leans over to listen to his cat. As he does, his right hand is at seat level. A second later, his right hand is up by his shoulder. (00:22:20)
Continuity mistake: In most shots, the Constellation's impulse engines are lit up orange, but when moving to meet the Enterprise, they're not lit up, though the ship is moving.
Continuity mistake: Spock nears the command chair, where a seated Decker is leaning on the left armrest. A second later he is leaning to the right.
Revealing mistake: As Charlie watches the pole lowering into a square hole in the deck, the wood grain of the set piece is visible. The wood has been painted gray/silver to look like metal. (00:06:06)
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 the original version, the planet is the same as the one used in "Bread and Circuses", but when the Enterprise leaves orbit at the end, it is a totally different planet, as it is the one used in "Friday's Child".
Continuity mistake: When the Enterprise is trying to push away the asteroid using the deflector, a quick shot of Uhura is shown from stock footage of "And the Children Shall Lead," but she is not in the rest of the episode.
Tomorrow is Yesterday - S1-E20
Other mistake: The Air Force police officer (played by Hal Lynch) was beamed up from inside the records room (after finding Kirk exiting the dark-room) but was returned outside the room... so he was physically transported INTO himself (as opposed to replacing the beamed-up future version at the exact moment and place he was removed as they did the pilot, Christopher).
The City on the Edge of Forever - S1-E29
Factual error: When Edith Keeler is killed by the truck, it is traveling at about 25 miles per hour. At that speed and her weight around 120 pounds, she would have been thrown about 50 feet. However, she's seen dead on the ground in front of the mission where she was hit.
The Conscience of the King - S1-E14
Plot hole: After Kirk sees Leighton laying next to the rocks, he rushes over to examine him. He places one hand on Leighton's arm and the other on Leighton's waist and declares "It's Tom Leighton...he's dead." All this without the use of a medical tricorder or even the experienced opinion of the ship's doctor to confirm what Kirk is seeing. (00:10:46)
Other mistake: The only way to beam up to the ship is to repeat the chess counter move after ASKING to be beamed aboard. After the two Captain Kirks fight ends with Spock stunning Garth with the phaser, Spock contacts the Enterprise. After saying "Spock to Enterprise, and hearing Scotty say back to him "Scott here", Scotty says" Queen to Queen's level 3" without Spock requesting to be beamed aboard or even waiting to see what Spock wanted in the first place. (00:46:56)
Other mistake: Stiles gives a countdown for impact to Kirk saying "estimate it will overtake us in two minutes, sir." After only 31 seconds, Sulu starts his own countdown, saying "5, 4, 3, 2, 1, impact." Only 55 seconds have passed from the time Stiles started his countdown to actual impact. Everything was shot in real time without scene changes. Being off the impact estimate time by over a minute doesn't sit well considering a Starship's advanced sensors. (00:29:27)
Continuity mistake: Kirk turns his head and displays a stunned look... a full second before the Gorn even starts to disappear. (00:45:47)
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.