When Kirk materializes in the transporter room, he is standing on a pad near the front of the chamber (near the steps). A subsequent wide shot from within the transporter chamber, establishing there to be no one at the transporter controls, also has Kirk's transporter pad empty. Then we return to the angle on Kirk and he is standing on the same pad again.
The positions of Uhura's arms change between shots when she is describing getting the runaround from Starfleet and the Federation.
At 39:31, the man to Ambassador Hodin's left turns his head toward Kirk and the "skin" behind his ear distorts, revealing that the actor is wearing a bald cap.
During one conversation between Spock and the Gideon Council, we get alternating shots of Spock, then a reverse angle to the view-screen with Ambassador Hodin's face. At the end of the conversation, the image on the view-screen changes to give Spock a closeup of Hodin's finger moving to the button to disconnect the transmission, then another change as we see Hodin's face fade out. Why would the image on the view-screen change?
Spock is told by the leader of Gideon that, after an extensive search, they confirmed that Kirk was not anywhere on Gideon. Later he said that "only one representative was allowed". Yet, Spock did not catch the fallacy.
When Kirk first arrived on the fake Enterprise they were in orbit. Later, they are no longer in orbit and Kirk never took the ship out of orbit. He did not bother to check with the computer to see why they left orbit.
When Spock confronts the Gideon guards, the second guard swings at him--obviously missing his face, in spite of the sound effect.
Of all of the methods offered to solve Gideon's problem of overpopulation, the most obvious solution is never addressed: sending the surplus population to colonize other planets, something done regularly by everyone else in the Trek Universe.
If the Gideons have no moving space, how do they keep their infrastructure, basic utilities, government (let alone the spacious council chamber we see), etc., functioning at all?
On an overcrowded planet where every surface is covered by people rubbing up against one another, their desperate concern is finding alone-time rather than food. How they derive sustenance, and contend with body wastes, are secrets of galactic benefit worth ferreting out, yet nobody considers it.
Both Dr. McCoy and Mr. Scott know what Kirk said before he beamed down to Gideon, even though McCoy was not in the transporter room to hear the comment.
After Captain Kirk beams aboard what he believes to be an unmanned Enterprise drifting in space, he makes absolutely no effort to ascertain the ship's location, navigate the ship, ask the computer any questions, or use his communicator to contact Starfleet or anyone else. Instead, he wanders around the ship, comforts a strange woman who is aboard the ship, and claims to know where the ship is located simply by looking at the stars visible on the main viewer (even though he knows someone or may be manipulating his setting).
It is plausible that Kirk could be confused for a bit by the fake Enterprise, but his secret codes on his safe or the secret codes needed for the computer could not have been known by the leaders of Gideon. He should have figured out quickly that he was on a fake ship.
When Kirk first beams back onto the deserted Enterprise, and he calls throughout the ship looking for personnel, it shows the Sick Bay is in "Red Alert" although the rest of the ship is not.