Charlene Masters is addressed as "Lieutenant" several times, but her tunic has the braidless sleeves of an Ensign. Moreover, she wears a blue sciences tunic, but she works in engineering, which involves ship's services (red tunic) personnel.
At the very beginning when Spock is reporting the atmospheric composition of the planet he has scanned he says "Oxygen Hydrogen atmosphere". This would be a very explosive combination in the right quantities (Hydrogen + Oxygen = Water + a lot of light and heat!). I suspect he meant to say "Oxygen Nitrogen atmosphere" (similar to Earth) but just got his line wrong.
In the second sickbay scene, when Lazarus is on the bed, the amount and shape of the blood around Lazarus' mouth changes in each shot, alternately getting larger, then smaller, then larger, then smaller again, all within the space of less than two minutes.
When Lazarus leaves his bed in the sickbay at around the 30 minute mark, the bed continues to broadcast his heartbeat. Indeed, when he starts to have an attack, his heartbeat speeds up. Normally, the sickbay bed can only read a person when he is lying on it; the sound should have stopped as soon as he stood up.
After scanning the planet, Spock declares that it contains no life. The subsequent appearance of a singular humanoid caused Kirk, Spock, and a security team to beam down to the planet to investigate. They do find a humanoid; but there are also trees, bushes, and grass on the planet.
Where did these plants forms of life come from, and why didn't the ship's sensors detect them? Moreover, why isn't Spock or anyone else surprised by their existence?
When the code factor one transmission comes through from Starfleet Command, before Commodore Barstow appears on screen or is otherwise identified, the bridge screen is still showing the planet below, and while the planet below is still showing, Kirk says "Kirk here, Enterprise standing by Commodore," although Kirk would have had no idea whether he was about to be speaking with a Commodore or an Admiral.
When the fire starts in engineering, and the assistant engineer drags Masters out of the room, when they are almost to the door the smoke in the room is still relatively light, and there is not enough of it to obscure vision, but less than 3 seconds later when the door to the corridor opens, the smoke is so thick that it is not possible to see through it, a solid wall of smoke.
The main threat in the episode is flawed. While in reality matter and anti-matter coming into contact with each other WILL cause a tremendous explosion (both the matter and anti-matter convert instantly into energy with 100% efficiency) the particles do not need to be identical as stated. In fact, the instant that Alternate (i.e. anti-matter) Lazarus came into contact with ANY matter including atoms of gas in the air, the explosion would have occurred but while it would have destroyed the "Enterprise" and possibly the planet, it would have no effect on the rest of the universe and none on the anti-matter universe (although the matter Lazarus would have a similar impact there).
It is highly unlikely that the Fahrenheit scale would be used in 23rd Century on a starship. Even now here on Earth in the 21st Century the use of Fahrenheit has been dropped from virtually all countries (the largest of those remaining being the USA) and even then its support is set to die out over the next 50 years far less 200 years! Spock would more likely use the Kelvin scale if anything when on a space vessel and describing science based measurements.
Lazarus never introduces himself by name on screen to Kirk, yet Kirk calls him by name during the first planet search. We come into Kirk's interrogation of Lazarus part way through. There is no way we can know exactly when Kirk learned Lazarus' name.
After Kirk returns from the "Anti-matter universe" he approaches Lazarus from behind and grabs him in order to force him into his ship so that he will become trapped in the corridor linking the two universes. During this confrontation the two security officers are clearly seen and are just standing there watching. Kirk glances over his shoulder and commands them to "Stay back!" but neither guard has moved at all.
The goof item below may give away important plot points.
There are two separate thefts of dilithium crystals. Each theft takes two crystals for one of the two timeships - one of which is in a different parallel universe. The crystals in "our" universe are recovered after Kirk's return. The question is: How did they recover the crystals in the other timeship, in the parallel universe? Also, being an antimatter universe, how would dilithium crystals from our universe work properly?
Kirk says that Lazarus will remain locked in combat with himself "forever" and through "eternity," and Spock agrees with him. For this to be true, Lazarus would have to be an immortal being, yet this story has clearly shown this not to be the case, as McCoy has treated Lazarus for an injury and found him to be an ordinary humanoid. With this in mind, Lazarus would only be locked in combat until one or both of his forms collapsed from injury or exhaustion, which should occur within a day. Where Kirk and Spock reached the "eternity" conclusion is never explained.
If the 'winking-out' effect is felt throughout the galaxy, why was it never noticed before? Anti-Lazarus refers to his people having learned to come through the portal to our universe, and that knowledge of the existence of the anti-universe drove our Lazarus insane. The implication is that the portal has been in use for quite a while. If so, the 'winking-out' effect would have been occurring for quite a while, and Star Fleet would have long since investigated and dealt with this situation.