When Kirk and Chapel first beam down to the surface, they are standing apart on the pads in the transporter room. When they materialize, they are right next to each other, nearly touching.
As Brown succumbs to Kirk's phaser shot there is no damage, not even any discoloration of the fabric of his jumpsuit. The catastrophic abdominal injury appears in a subsequent take.
When Kirk and Nurse Chapel first beam down to Exo III, you can see Kirk's reflection on a glass screen behind the two characters. But the transparent screen is an intentional barrier between the corridor and the planet's frozen surface, which can be seen when the camera pulls away and one sees 4 straight sides and 4 corners.
One close-up of Kirk during his last conversation with Ruk is flipped. Notice that his hair is parted the opposite way and the colors of the costume are reversed.
In the close up shot of Kirk's hands clinging to the cliff, the cliff bends like rubber.
When Kirk pulls the stalactite from the roof, it does not snap like limestone, but bends like polystyrene. However, the structures on another planet may not be made of limestone at all. They may be organic.
The rope Kirk unties to escape doesn't actually tie him down to prevent him from doing anything.
When Christine chases after Ruk while he is chasing Kirk, she reminds him that Korby told Ruk he has to obey her commands. But she was not in the room when Korby told Ruk this.
On the android turntable, Kirk verbalizes his message to Spock "Mind your own business Mr. Spock, I'm sick of your half-breed interference, you hear?" The three androids producing android-Kirk would have heard the message and aborted the procedure.
When Roger threatens Kirk with Kirk's Type I phaser, he's holding it upside-down.