As has been pointed out in other episodes, Spock has super-sensitive hearing, yet he is completely unaware of the the conversation between Riley and Sulu when Sulu decides to leave the Bridge. Even when Riley nervously looks over to Spock's station and then says "Sulu" loudly enough to be heard with normal hearing, Spock doesn't hear him or know that Sulu has left the Bridge until later.
When the first away team beams on board Scotty calls Kirk to let him know that they are being decontaminated. Kirk says "you better have medicine check them out." Medicine is in reference to pharmaceuticals; Medical would be a place on a ship.
As a bridge officer, Uhura no doubt was trained to assume the navigation station in an emergency. However, It is doubtful that a Yeoman such as Rand would be qualified to "TAKE THE HELM!!" when Kirk orders it.
When Spock enters the briefing room after being infected, the "Briefing Room 2" sign hangs on the bulkhead to the left of the door. When Kirk exits the same briefing room later, the sign is now hanging to the right of the door.
When Kirk and Scotty break into the engine room, Kirk grabs the back of Riley's chair and turns it around to face him. There is an immediate cut to a close-up frontal shot of Riley, and Kirk's arm is nowhere to be seen on the chair.
When Scotty finishes cutting through the bulkhead in order to regain entry to Engineering, the piece he cut out is very hot and in fact is smoking, but Scotty grabs it with his bare hands to remove it and yet doesn't get burned. Update: Note that Scotty grabs the cut panel at the bottom-right, where he had cut it much earlier. The panel was only smoking from the top, where he had just finished cutting it. The panel would have had time to cool at the bottom-right by the time he grabbed it. And, given the location of the cut panel, and the equipment behind it, the material used in the bulkhead would almost certainly have properties that would insulate the crew from any heat generated by the equipment behind the panel, so the cut piece would have been cool enough to handle.
When Tormolen dies on the operating table, the body temperature gauge drops from normal to 90 degrees Fahrenheit (and presumedly lower) in a few seconds. However, normal bodies cool at just 1-2 degrees an *hour*. (This occurs many times throughout the series; this is the first death at a surgical table.)
When Kirk specifies a course to leave the planet, he specifies a 'hyperbolic' course, which makes no sense, since it refers to a hyperbola. A hyperbola can be thought of as two parabolas facing each other (their open ends pointing away from each other). How that figure can be used to define the course of a single ship is difficult to imagine. It would be more likely that the correct course description should have been 'parabolic'.
Granted that after Lieutenant Riley turned the ship's engines off, it would have been impossible to re-start the warp engines in the time available except by using risky method depicted. However, there was no mention of whether it was possible to re-start the impulse engines, which are capable of all speeds within the sub-light realm and would certainly have been more than sufficient to restore the 'Enterprise's' orbit.
Kirk's log entry at the episode's opening makes no mention of anything being amiss at the station below. However, Spock and Tormolen are both wearing thermal suits when they beam down inside the structure. They obviously knew something was wrong, but why was there no mention of this in Kirk's log entry?
When McCoy goes to inoculate Kirk with the vaccine/serum, he rips the shirt off Kirk's shoulder before using the hypo. This is the only time this occurs when a hypo is used, as it is normally established that hypos work through clothing.
There seems to be no rhyme or reason to how the infection is spread. Sulu and Riley are infected from fighting with a fully-clothed Joe Tormolen (but they did have contact with Tormolen's blood after he stabbed himself), and Chapel by a calm Riley, but Uhura, despite being manhandled by a sweating, stripped-to-the-waist Sulu, is not infected. People's immune systems and tolerance levels vary.
After it becomes clear that Riley is infected, Spock orders him to report to Sickbay and then calls security and tells them to "see to it that he arrives". Although Riley does go to Sickbay, he doesn't stay there and ends up taking over Engineering. Security apparently never follows up on Spock's order. If they had, they would have been there to stop Riley before he could get to Engineering.
Towards the end of the episode, Kirk orders a hyperbolic course, and Lieutenant Brent, at the navigator's station, asks for a direction. But he turns to the right to face Spock's station (Spock is in engineering) when he asks it. Kirk is over Brent's left shoulder, but if the actor Frank da Vinci had turned that way to deliver his line, he'd have had his back to the camera.
The goof item below may give away important plot points.
After the warp engines implode, the ship is going backward in
time. They reverse course and time flows forward. The ship's chronometer is calibrated wrong - it goes from 7:59 to 7:00 to 8:01. (This has been corrected in the Remastered Edition.)