When Lincoln first appears on the Enterprise's viewscreen, Uhura has a different hairstyle in a closeup than she does in the rest of the episode.
When Mr. Lincoln beams aboard, there is a large view screen on the wall next to the scanner behind Scotty. When he, Kirk and Spock beam down to the planet a short time later, the screen is gone.
When Kirk and Spock are making spears and Surak yells for help, there are limbs on Kirk's spear, then, a few frames later, it has none.
Obvious stunt double for Kahless in the big melee toward the end. Stuntman is much larger, with paler skin.
During the first fistfight between the Enterprise crew and the evil characters, Kirk is struggling with Col. Green. Kirk is knocked on his back, and as he draws his legs to his chest to kick out at the colonel, his pants appear to split - a white line appears on the seat of Kirk's uniform pants (further evidence of William Shatner's weight gain).
The makeup and/or glue for Lincoln's fake beard is seen several times, most notably under his lower lip.
In the briefing room, as Kirk is talking to McCoy and Scotty before beaming down, a script or newspaper on the conference table is shown in reflection on the three-sided viewer.
At 25:52 (both on the original version and the version with enhanced special effects and audio) the image of Captain Kirk speaking to the molten creature is backward. This is extremely noticeable as the Star Fleet symbol on his shirt is suddenly on the right side instead of the left side. This is surprising that this wasn't fixed this when it was being remastering for 1080P. It is extremely hard not to notice.
When Lincoln is crawling en route to save Surak, he's wearing elastic-sided shoes typical of the Enterprise crew uniform, rather than 19th century shoes.
The entire premise of this episode is highly flawed. The aliens set up the whole battle in order to learn about good and evil, which were unknown concepts to them; but aside from Kirk and Spock, the ones doing the fighting were fabrications who behaved exactly as good and evil would have behaved, so they must have known about both concepts in order to have them behave in the correct manner.
The creature tells Kirk that, unless he agrees to the battle, the Enterprise and crew will be destroyed within 4 hours. Suddenly it's 2 hours later and Kirk and Spock haven't done anything to prepare - no weapons, no defensive strategies or places found to take cover. But Kirk does agree that they are running out of time.
Kirk tells everyone that Lincoln will be treated with presidential honors before he is beamed aboard the Enterprise. After Lincoln has materialized, a couple of guards are pointing their phasers at him, which would not be even close to presidential honors.
The rock creature uses fake leaders from Vulcan and Earth's past to side with what they call good. There are four others created to side with what they call evil. The test is invalid and lacks scientific method. They can not validate either side because six of the characters in the play (as they called it) are fabricated from Kirk and Spock's subjective views of reading history.
The aliens want to see how the crew represents good vs evil, but the embodiments imitating Surak and Lincoln do all the decision-making.
Mr. Lincoln asks Captain Kirk, "Do you still measure time in minutes?", to which Kirk replies, "We can convert to it, sir." Hours and minutes are used regularly in the Trek Universe.
TOS was notoriously inconsistent in references as to when the series takes place. This one story contradicts itself on the matter. Early on, Scotty says that Abraham Lincoln has been dead for 3 centuries, suggesting that they are in the middle of the 22nd century. Later, Colonel Phillip Green is said to be from the 21st century and that his treachery happened "centuries ago." The plural reference to centuries since the 21st suggests they're past the 22nd and into at least the 23rd.
Surak tells Kirk and Spock that he does not believe in violence and refuses to fight. However, he has already done so in fighting off one of the four evil opponents in their initial attack.
Lincoln says, "I issued orders that sent 100,000 men to their deaths at the hand of their brothers." The actual number of deaths resulting from the American Civil War was over 600,000.
In the opening shot of the bridge with Kirk in his captain's chair, you can see William Shatner looking at the camera filming him from his right before he turns his eyes to the main view screen.