Kirk may possibly violating the "Non- Interference Directive" by persuading Spock to take command of the Enterprise and make changes. The Federation is not supposed to interfere with the ways of another alien race.
In beginning scene when Kirk and the others first beam back up to the Enterprise, Kirk signals the ship to engage, then puts his communicator away and puts his hands down by his side. As they first appear in the transporter pad aboard the ship, he has his communicator still in his hand, held up to his mouth. During the original series, movement during transport was not possible.
When the landing party is in the middle of the cross-universe transfer, the USS/ISS Enterprise is seen in orbit going in opposite directions (USS Enterprise going counterclockwise [when viewed from above]; ISS Enterprise going clockwise). However, through the rest of this episode, the ISS Enterprise is going counterclockwise.
In the first assassination attempt by mirror Chekov, the guard in the blue suit that turns and breaks up the assassination is first located by the turbolift doors (to Kirk's left) when he shoots the first red-shirt, then is suddenly at Kirk's right-hand side when he shoots the second red-shirt.
The mirror universe has the Terran Empire, not the United Federation of Planets. Yet about halfway through the show, the mirror-universe characters begin to refer to Starfleet and the Federation and don't mention the Empire again.
The "mirror" versions of the Halkans behave no differently than their counterparts in the "prime" or original Star Trek universe. Both the "prime" and "mirror" versions of the Halkans are pacifists and refuse to cooperate or submit to the Federation or the Terran Empire respectively.
The communicator scenes between Kirk and Scott following Spock informing Kirk of his orders to kill him. Scott is obviously reading his lines from somewhere off screen. Notice in all his deliveries he is looking at a fixed point to his right.
The goof items below may give away important plot points.
As Mirror Spock prepares to transport the landing party back to its rightful universe, he tells Capt. Kirk, "You have two minutes and 10 seconds." As Kirk tries to convince Mirror Spock that both he and his empire are "illogical," Mirror Spock tells him, "You have one minute and 23 seconds," meaning that 47 seconds have passed. But only about 30 seconds have actually elapsed.
When Chekov enters the bridge turbolift just before Kirk (just before his assassination attempt), his shadow can be seen from the standard bridge camera view, indicating a bright light source to the left of the turbolift car and a missing wall.