In the teaser the Enterprise banks like an aeroplane (not for the first time). In space this would not happen as manoeuvrability is not dependent on air flow.
The Enterprise crew are shocked to learn that their surprise capture may be due to a new cloaking technology possessed by the Romulans. Kirk and crew already encountered cloaked Romulans two years earlier in Balance of Terror (1966). However, the cloaking technology in The Enterprise Incident is new because it is improved. The Enterprise was still able to track a Romulan ship, though not accurately, in 'Balance of Terror' and cannot detect them at all with this improved version.
Spock reports that the Enterprise is surrounded because there are Romulan ships ahead and behind them in the same plane. This is two-dimensional thinking as there are no ships above or below them. However, they are effectively surrounded because the Romulan ships can fire above and below the Enterprise.
It seems strange that humans know that there is no such thing as a "Vulcan death grip", but that the Romulans, who are said to be the offshoots of Vulcans wouldn't have that knowledge. However, it is never stated that humans, suggesting "All Humans", know about the Vulcan Death Grip. Nurse Chapel mentions it. Not only is she a medical officer who would know all crew members physical strengths and weaknesses, but she also has a special interest in Mr. Spock.
Around 7 min, when Kirk is debating what to do about the situation, Scotty says, "If the Enterprise is taken by the Romulans" his mouth movements do not match what he is saying.
Transporter beams cannot penetrate shields, yet Kirk is able to beam to the Romulan flagship and back. This means that the Enterprise and the Romulan warships have their shields down, an unlikely scenario with hostile warships facing one another.
After Liviana leaves Spock alone, he pulls out his communicator to contact Kirk. As a non-Romulan, it makes no sense that Liviana would allow him to keep his communicator - even if he is somewhat trusted by her.
Deck 2 is only one deck below the main bridge, yet when Spock takes the Romulan Commander there in the turbo lift, the journey takes an inordinate amount of time, longer than most trips shown to places like deck 5 or main engineering.
When Scott leaves the bridge and entered sick bay, he immediately sees Captain Kirk after plastic surgery alters him to look like a Romulan. If this was supposed to be secret, why was he so near the sick bay entrance? Anyone could have walked in and easily discovered Captain Kirk in his altered state.
When the Romulans contact the Enterprise the second time, Kirk has it put through to the conference room. Kirk looks down to the three-sided video screen and starts talking to the Romulan ship but there is no screen facing him. One of the three corners of the screen is facing him.
In the early second scene in the briefing room, where Spock reveals to the Enterprise officers that the Neutral Zone was breached on Kirk's orders, two officers are shown in the background. One is wearing a gold tunic indicating he is a member of Command or Navigation with lieutenant commander insignia on the sleeve (holding the same rank as Dr. McCoy and Mr. Scott and outranking both Chekov and Sulu.) Another is a "red shirt" indicating he is a member of Security, Engineering, or Communications and that contains stripes indicating a (full) Cmdr. ranking (the same as Spock) which would in turn indicate that he outranks both Lt. Uhura and Mr. Scott! No explanation or other identification is provided.