When McCoy falls onto a blue rug, the position of his head in relation to the edge of the rug keeps changing between shots.
When Natira opens the door to the Oracle, she passes her hands one at a time in front of panels attached to both sides of the door. Later, when Spock repeats her actions, he waves one hand in front of a panel and the other across the wall between the door and the panel. Despite this, the door still opens.
When preparing to go to Yonada, Kirk enters the transporter but his right foot is off the disk on the floor when the camera cuts and we hear the transporter activation sound. When he arrives he is fully intact, foot and all. However, this is a common practice, as they are able to transport large crates of equipment which is much bigger than an individual pad, which shows one does not have to be exactly on the pad for transport.
When briefing Captain Kirk, Mr. Spock says that the asteroid is "200 miles in diameter", nowhere near the size of a planet so it can't have enough natural gravity for humans to walk normally as depicted. The more so as the object is hollow. To the same token, the gravity force would be more or less centered in it and there would be no normal "top" and "bottom" as shown. But it is highly likely there is an artificial gravity system aboard.
When Spock is thumbing through the Book of the Fabrini, although Kirk asks whether it is indexed, and Spock says yes, in fact all pages can clearly be seen to be blank until he reaches the pages he wants.
When he is transporting to the asteroid, a purple blob appears on the right half of McCoy's tunic as he materializes, then abruptly disappears. This is likely due to film damage when the materialization sparkle was added in post-production.
The walls in the oracle room are decorated with triangular panels, every second panel being upside down, bearing letters/words of the Yonada language. Those "hieroglyphics" are the same on every panel, supposing that the alphabet of the Yonadians seems to be very restricted despite their huge scientific knowledge.
The footage of Kirk arriving on the bridge and Spock briefing him on the missiles is repeated when Kirk gives his Captain's Log after the opening credits.
It's never explained who or why someone or something from the asteroid Yonada fired missiles at the Enterprise in the first place.
When the landing party first sees the cylindrical entrances at 07:32, stage crew is visible moving behind the rocks on the extreme right.
When Kirk stands on the altar and presses the center, triggering the entire altar to slide out from the rear wall, the shadow of a stagehand can be seen behind the altar. The stagehand physically slides the altar and the shadow of his arm can be seen to lower, once he stops pushing the altar away from the wall.
When Kirk and Spock are first punished by the Oracle, they are almost electrocuted, but when they are about to enter the Oracle's control room and disable it, all it does to stop them is increase the room temperature.
Early on Natira said "When we get to the new world", yet later she doesn't want to believe that she's on a ship. How is it going to get to a "new world", if it's NOT a ship? If it's a planet that moves, it's a spacing going vessel.
Kirk orders McCoy to come back with him to the ship. But Kirk had already put in for a McCoy's replacement, so if McCoy had stayed the Enterprise would have not been without a doctor.
Spock says the asteroid is "200 miles in diameter" rather than using Federation standard metric units (kilometres).