At the end, Captain Kirk tells Mr. Spock, "Full ahead, warp factor one" which Spock repeats back to him, however, Mr. Spock is sitting at the science station (his normal station) and not at the helm, which is where the ship's movements are controlled. While Spock is not at the helm, this exchange is reflective of Naval command structure where the Captain tells the First Officer what he wants done and the First Officer orders the Crew. This was not the normal for Star Trek The Original Series as it progressed, but it reappeared in Star Trek: The Next Generation with Number One giving the orders.
At the very beginning Kirk tells Spock they are "hundreds of light years from Earth" and then after the opening credits Kirk says "in the distant reaches of our galaxy..." Our Galaxy is approximately 100,000 light years in diameter so even "hundreds of light years" would still be relatively next-door rather than "distant reaches"
Near the end, when Spock and McCoy are debating on how much longer to wait for the Captain before testing the antidote, the scene is cast in a reddish tint. When Spock is walking out of the lab, the red tint abruptly changes to normal lighting.
The children abduct Yeoman Rand and tie her to a chair in their classroom/meeting room. Three loops of heavy rope, wound around her torso, secure her to the backrest of the chair. What's odd is, in the side-view shots, the ropes are looped UNDER her elbows; but in the frontal shots, the ropes are looped OVER her elbows.
Notice the lit Bunsen Burner (at around 10 mins). One could expect that the Gas Main would not still be going after 300 Years. Bunsen Burners weren't used in the labs on the Enterprise, however the Enterprise could have transported equipment down for them. It is also possible that they could have used a different source - such as propane, butane or white gas for the Bunsen Burners.
Given that the Captain's Log stated that the SOS Transmitter was found in the same building as the Lab, why couldn't someone (probably Mr. Spock) have used this to signal the Enterprise to beam down some more Communicators, thus solving their dilemma? He will make a computer out of local items from an even earlier time period in Star Trek: The City on the Edge of Forever. Also, when the children steal the communicators, they only get three - presumably Kirk's, Spock's, and McCoy's. Rand and the two security men are not present, and therefore probably have their communicators with them - it is highly unlikely Kirk would let them go off into unknown territory without standard equipment.
What did the two security guards that came with the landing party do? They aren't present when the away team is attacked or in any other scene, did they go AWOL? We see them again at the end and they aren't sick with the purple sores, either, when they return. In addition, they would have communicators with them; John only gets four communicators when he raids the lab, so the landing party would be able to communicate with the ship using the guards' communicators.
"Miri" established the Enterprise's strict zero-tolerance quarantine protocol, which was promptly abandoned and never mentioned again - in the rest of the series they beam aboard infected people left and right. Since reaction to "Miri" was very negative, it's likely that the writers decided to simply ignore it rather than work it into continuity.
It is established that the children do nothing but play all day, but it has been 300 years since any non-infected adult has lived, so what do the children eat? The landing party makes frequent references to the children's food sources running out soon, but there was no technology in 1960s Earth (which this planet's civilization was identical to) that could have kept foodstuffs edible for this long, so it's unlikely children would have survived this long without starving to death or at least developing severe vitamin deficiency.
One of the dilemmas faced is the planet's dwindling food supply. It is unclear how a society of such children had been able to cultivate, amass, maintain and sustain a food and water supply for a period of approximately 300 years. Similarly, much of the clothing worn by the kids appeared well maintained for such a period of time, and there's no explanation as to how well most of the kids hair was cut, considering the lack of any adult presence.
On a planet the size and with continents comparable to Earth, the Enterprise encounters the population of only one city. Presumably the plague effected the planet's entire population, otherwise adults from other areas of the world likely would have responded to the children's SOS. It is unlikely the entire population of survivors managed to all migrate and settle into one area, and it is unclear if the Enterprise was able to seek out other areas and cure the planet's entire population.
In the opening scene, where the ENTERPRISE orbits Miri's planet, you can see the hole at the top of the globe used for the original version of the scene where it would have been attached to its stand. This has been replaced in the remastered version with a digital shot of that same planet.