The ape-like creature is referred to throughout as a "Mugato," but in the closing credits it's "The Gumato" (see trivia).
At the beginning, McCoy's equipment changes position relative to his feet as he, Kirk and Spock beam up to the ship.
When Nona confronts the villagers with the phaser, the man in the light red leans on his flintlock with both hands. During the next shot from behind he is holding the rifle with one hand, and then back to leaning with two hands.
After Spock is shot and is falling to the ground, there is no gunshot wound on his back, and no blood is visible. When Kirk reaches Spock, there is suddenly a great deal of green blood on both the front and back of his shirt, but there's still no apparent entry wound, as the shirt isn't ripped or torn in any way.
However, most movies and TV shows from this era were reluctant (partly due to censorship) to show the effect of gunshot wounds realistically.
After Spock is shot and they rendezvous with McCoy and beam out, there is an error in the 'optical.' Not all of the equipment 'sparkles' with the beam out - one piece just dissolves, clearly missed by the technicians in post-production.
Obvious double for Spock when he's wounded by gunshot.
Obvious stunt double for the Klingon (Krell) when Kirk attacks him in the forge room (possibly the same man that doubled for Spock when he was shot at the beginning).
The Mugatu tracks are too small and the wrong shape to be made by the it. This is because stock footage of the White Rabbit's footprints from Shore Leave (1966) was used for the Mugato prints.
When Tyree picks up a rock to brain his enemy, he clearly strikes the ground beyond the man's head.
Tire tracks are visible as the villagers climb the hill while pursuing Kirk and Spock.
When a villager grabs Nona from behind, some San Fernando Valley buildings can be seen in the background.
Notice when the lock of the flintlock is shown closely, there is no flint in the hammer. Therefore no spark would be made and the flintlock would not have fired. The sound made by the firearms sounds more like M1 Garands than muskets.
At the end, Kirk retrieves his communicator to answer the ship, but it never beeped to indicate an incoming signal.
In the opening scene, after Kirk distracts the Villagers by throwing a rock at them, the Villagers begin to chase Kirk and Spock while firing the guns at them. McCoy is nearby and hears the shots fired. He then stands and runs a few steps to his right, while the camera pans to follow him. When he stops, there is a small pedestal microphone visible in the lower-left part of the screen, and it stays there the entire time he is communicating with the ship.
In the opening sequence, Kirk and Spock investigate a shout (by a Villager) of 'Take cover here!' Given Kirk and Spock's location at the time (behind a large rock formation), the group of Hill People would also have heard the shout and been alerted to the Villagers' presence.
Scotty says that a flintlock would be the first firearm a society would develop. Actually, the matchlock and wheel lock predate the flintlock.
When explaining how the flintlock works, Kirk says the hammer striking the pan causes sparks. In reality, the hammer striking the frizzen (part of the cover of the pan) creates sparks which fall into the pan.
Kirk states that he was wounded by the Gumato's claws. Later, however, it is stated he had been bitten by it.