Factual error: Kirk, McCoy, and Spock are chained to the dungeon wall, with McCoy noticing he is next to a skeleton, also chained. This is incorrect, as a skeleton's bones are held together by cartilage, which would rot over time. Therefore, the skeleton would not be hanging intact, but would be a pile of bones on the floor.
Suggested correction: As we learn in the episode, the 2 aliens created the entire castle only recently to appeal to the human's primal fears and superstitions. A pile of bones on the floor would not have the frightening impact of a hanging skeleton.
The City on the Edge of Forever - S1-E29
Factual error: When Edith Keeler is killed by the truck, it is traveling at about 25 miles per hour. At that speed and her weight around 120 pounds, she would have been thrown about 50 feet. However, she's seen dead on the ground in front of the mission where she was hit.
Requiem for Methuselah - S3-E19
Factual error: Spock plays a piece on a harpsichord that he says is by Brahms, but Brahms was a late romantic composer and the piece is a simple baroque dance piece. Also by the time of Brahms the harpsichord was already obsolete, a composition like this wouldn't be sitting on a harpsichord.
Suggested correction: The first sentence is logical; if Spock is able to recognize the style as Brahms, then it should not possess the style and structure of Baroque music. The second sentence is not necessarily true because some romantic composers did write for the harpsichord. For instance, the late romantic composer Richard Strauss composed, "Divertimento for Chamber Orchestra after Keyboard Pieces by Couperin", which is scored with a harpsichord part.
Factual error: In the opening sequence, Spock identifies the alien ship as possessing "ion propulsion" which he says is "unique technology." Scotty is similarly impressed and says, "They could teach us a thing or two!" Kirk later comments that "Advanced ion propulsion is beyond even our capabilities." However, even back in the 1960s, ion propulsion was physically feasible, while Warp propulsion was complete fantasy. Ion propulsion of any kind could never even reach lightspeed and would be incredibly primitive compared to Warp technology. In fact, we in the 21st Century have already developed ion propulsion, but it will probably take many hundreds or thousands of years to develop anything even close to Warp technology.
Factual error: If this planet is a duplicate, an exact copy, of Earth, then Lake Okeechobee, the largest lake in Florida, is wrong. Seen from space, it is much larger and more distinct than the lake shown on this duplicate.