When Uhura's rescue party assembles in the transporter, Chapel's uniform is blue. When they materialize on the planet, it is initially red, then blue in all subsequent shots, even though a few shots later one of her sleeves turns red.
As Spock lies in Sick Bay giving instructions to Nurse Chapel, a nearby female crew member changes from African-American (in wide shots) to Caucasian (in close-up shots of Spock).
At 12:09, McCoy and Carver are seen not wearing their headbands. In all subsequent shots, the headbands are there.
Theela's earrings in most scenes look like crescent moons, but when she first appears and when she first talks with Uhura, her right earring is triangular.
For the urn to fill with water from rain falling into it (thereby drowning the men hiding inside), there would have to be a massive rainstorm with many feet of water, not the ordinary storm that is depicted. On the other hand, the men are weakened; they could easily have drowned in less than a foot of water if they were on the bottom and unable to lift their heads (which is not how they were shown).
Spock reaches out telepathically to Christine Chapel. This ability was previously established in The Omega Glory (1968) in which Spock "makes a suggestion" to Sirah to open a communicator. Additionally, Lt. Uhura is right beside Christine but doesn't "hear" Spock. In Return to Tomorrow (1968), Spock's consciousness was briefly housed inside Nurse Chapel's mind. In her words, "We shared consciousness." This made it possible for her to receive his thoughts.
As Scotty records his log entry, the animation causes the planet background to twitch in its rotation.
When Theela greets the landing party, she addresses Kirk, Spock, and McCoy by name. However, she completely ignores Lt. Carver, the "red shirt" that is part of the landing party.
When Spock is rescued, he says nothing as to where the rest of the landing party is located. Why didn't Uhura ask him?
The Taurean women have been searching fruitlessly for the escaped men of the landing party. When Uhura beams down, she forces Theela to use her opto-aud to find the men immediately. Why didn't the planet women think of that, of using their own equipment to find the men (or was this a low-pitched sexist joke since the planet women were all platinum blondes)?
When confronted by Uhura, the leader of the siren females gives in and shows Uhura the past, using a device that they have a specific name for (opto-aud). Even though the leader never calls the device by its name, Uhura uses the name when asking if it can locate the men.