Continuity mistake: The Organians cause all the weapons to overheat, and everyone drops them to the floor. When Kor moves to the desk to call his ship, though, he's wearing his disruptor. It's gone again when he returns. (00:41:45)
Continuity mistake: When Kirk orders the crew to fire phasers; the exterior shots show the ship firing torpedoes, and torpedo explosions flashing out in space.
Continuity mistake: Kirk has a sunburn on his face and chest (it appears to end partway down his chest-most noticeable in the scene shortly after his identity is revealed to the Klingons). First of all, how does one get a sunburn while flying through space (and one that does not match the pattern of the uniform). If one wants to explain it away as being from a recent Shore Leave or Enterprise tanning bed, the issue still remains that the intensity of the sunburn changes as the episode continues.
The Alternative Factor - S1-E28
Continuity mistake: Both the positive and negative versions of Lazarus have a long mustache and a fairly full "waterfall" beard - until the scene in the briefing room, when inexplicably, the mustache and beard are both suddenly so thin they're barely visible. As soon as they beam down to the planet, however, the much thicker facial hair is restored. (00:23:00 - 00:24:30)
The Alternative Factor - S1-E28
Continuity mistake: Lazarus had knocked out the transporter technician, but when Kirk arrives he is fine and operating the transporter. (00:37:35)
The Alternative Factor - S1-E28
Continuity mistake: Mad Lazarus has just put a black-and-white hatch cover over the stolen dilithium crystal in his time ship. It's in place when Kirk arrives to confront him. But a moment later, when Kirk leans in and is accidentally transported to the negative universe, the cover has disappeared. It's still missing when Kirk returns to push Lazarus through the portal. (00:39:20)
The Alternative Factor - S1-E28
Continuity mistake: Kirk keeps referring to Masters as "Lieutenant", even though she has no braid on her sleeve - all Enterprise lieutenants have a single gold braid on each sleeve.
The Alternative Factor - S1-E28
Continuity mistake: Lazarus' goatee and facial injures are constantly changing.
The City on the Edge of Forever - S1-E29
Continuity mistake: While he and Kirk discuss the history tapes in their room, Spock's shirt collar is turned up on the left side in close-ups, but straightened out in the full shots. (00:32:55)
The City on the Edge of Forever - S1-E29
Continuity mistake: After Edith stumbles and nearly falls down the stairs, she still has both her shoes on. But after the kiss, she proceeds up the stairs and is now carrying one shoe. She had no time to take it off, as she was, er, busy with Kirk the whole time. (00:43:55)
The City on the Edge of Forever - S1-E29
Continuity mistake: When McCoy arrives in NYC while the man is stealing a bottle of milk, in the closeup the actor's ring finger is partially amputated, but when he steals McCoy's phaser his finger is intact in the closeup.
The City on the Edge of Forever - S1-E29
Continuity mistake: As Kirk and Spock prepare to enter the time portal, Spock deliberately places the tricorder at his left side, holding it there with his left hand. As they appear in 1930 earth, he is holding the tricorder with his right hand and his left hand is free.
The City on the Edge of Forever - S1-E29
Continuity mistake: At the end, Kirk says "Let's get the hell out of here." Captain Kirk and 6 others were beamed up at the same time, but the Enterprise transporter only has 6 pads; presumably a red-uniformed crew member was lost in space.
Suggested correction: Short version is, the Enterprise has more than one transporter room. The Animated Series references at least four on the Enterprise from this era. There are also cargo transporters, which aren't normally calibrated for personnel, but can be, albeit at reduced capacity and increased energy usage. https://memory-alpha.fandom.com/wiki/Transporter_room.
The City on the Edge of Forever - S1-E29
Continuity mistake: In the beginning, the ship is shaken, and McCoy is injected with cordrazine, and falls down. As he lays, the hypo is lying under his body. After the cut, he's on his knees, holding the hypo.
The City on the Edge of Forever - S1-E29
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.
The City on the Edge of Forever - S1-E29
Continuity mistake: When trying to catch McCoy, Kirk's pants get dirtied. After coming out of the Guardian's time portal in the Depression, his pants are clean.
Operation -- Annihilate! - S1-E30
Continuity mistake: After Kirk orders Uhura to reestablish contact with Aurelan's transmitter, when Uhura replies that they do not acknowledge her contact signal, Uhura's communications earpiece is not in her ear, but in the next shot it's suddenly back in her ear. (00:06:10)
Operation -- Annihilate! - S1-E30
Continuity mistake: When they put the satellites in orbit, Sulu announces they are in place. Then Kirk says "Energize" and the next shot shows the helmsman station and Sulu is absent - some guy in a red shirt is there instead. (00:45:30)
Operation -- Annihilate! - S1-E30
Continuity mistake: When Uhura tells Kirk that she's made contact with his private transmitter she's seated at the communications console, with the two overhead screens above her station, beside the turbo lift alcove. Yet after the contact's been broken, when she tells Kirk, "It's impossible at the moment because they stopped broadcasting," in this unusual shot facing Uhura the turbo lift alcove is not behind her - Uhura is actually seated much closer to Spock's station, note the upper section with overhead screens. Then she's properly seated at her station once again.
Continuity mistake: When Kirk confronts Spock in his quarters, Kirk asks if it's something only his planet can do for him, then Kirk uses his left hand to grab hold of Spock's arm as he drops the stylus, but in the closeup it's Kirk's right hand holding Spock's arm up, and that stylus is also back in Spock's raised hand. (00:12:25)