When Kirk and Spock break into Kor's quarters, Spock takes Kor's disruptor. However, later, when Kor grabs a knife, his disruptor is clearly seen in his holster. In the next frame, Kor's holster is empty again.
After Spock readies the grenade, he stars a countdown, saying "6, 5, 4, 3". He doesn't say "2, 1" out loud, but the explosion occurs at about 1 rather than 0. An error, for the normally mathematically-precise Vulcan.
Sulu is left in command of the Enterprise, even though Scott is on the ship (although not on screen) and should be the acting captain while Kirk and Spock are on Organia. However, given that Klingons are in the area, combat was a distinct possibility so Scott (or Kirk) presumably decided that the best place for him was Engineering, to oversee maintenance of weapons and engines.
The Klingon military decree is written in English. Since the "universal translator" only applies to speech, it would have been more interesting and consistent with Star Trek lore if the paper had been written in some alien script.
When Kirk is tossed into the cell and hits the "stone" floor, it's obviously a mat (to cushion the actor's fall).
When Kirk and Spock stun the first two Klingon guards, the one on the right can be seen bracing himself with his arms as he rolls down the stairs. The guard on the left jumps off the ledge instead of falling off.
When the Organians are reverting to their true form, they are so bright that none of the other characters can keep their eyes open looking towards them. However, Kirk's shadow on the floor is directed towards the supposedly painful bright light.
In the remastered version, the first encounter with the Klingons is with one D-7 class battle-cruiser, a ship that is, according to the canon, a match for an 'Enterprise'-class ship one-on-one. She shouldn't have been destroyed so quickly and seemingly easily. Later, when 'Enterprise' is in orbit around Organia, she is attacked by six such vessels. (Count 'em.) Such a force should have dispatched 'Enterprise' in short order.
During the initial Klingon attack, Kirk orders return fire from the Enterprise's phasers, but the ship is shown firing photon torpedoes.
When Spock was talking about the Organians being stagnant, he says that "there is no evidence of any progress as far back as [his] tricorder can register". Tricorders cannot see back in time.
In the end credits, George A. Rutter is listed as "Scpipt Supervisor".
If the Organians are omniscient, higher evolved beings they'd if not immediately stopping all warfare between sides (as they do later) even before the Klingons arrival then at least know Kirk and Spock were planning a violent act by exploding the munitions and interfere as they also do later.
After Spock drops a grenade in the crate of explosives, he voices a count down to the explosion. His lips never move.
During the initial Klingon attack, an unpainted piece of lumber from the set is directly below Spock.
Although Spock observes that Organia is a "a primitive society" which is "totally stagnant" and making no progress toward mechanization, neither Spock nor Kirk appears to notice that the doors to the Organian council chamber open and close by themselves when someone approaches. There is no evidence of any Organian opening or closing the doors.
When first describing the mind scanner, Kor tells what effect it would have on a Vulcan (Spock) and on a human (Baroner, who is really Kirk). At this point, Kor does not know that any humans are present. He thinks "Baroner" is an Organian.
When Kirk and Spock are walking down the street, Kirk accidentally bumps into a Klingon. Spock quickly defuses the tension. Afterwards, Kirk and Spock talk about Kirk's desire to act violently, and Spock refers to Kirk as "Captain" instead of "Baroner", his Organian alias. However, there are a few tables of Klingons at a cafe they are standing next to, who would've easily overheard their conversation about not being who they were pretending to be.
If the Organians have no use for violence at all, it is not clear why they have dungeons or cells (and quite capacious ones at that) into which Spock and Kirk are thrown.
If the crates Spock was planning on blowing up contained chemicals he could not obtain the sonic grenade from them.
When the Klingon officer informs Kor about Kirk and Spock's escape, we can see the boom microphone appear on the top right corner of the picture.
Near the beginning, Captain Kirk offers the Organians Federation assistance in a number of different ways, much of which would have been a clear violation of the Prime Directive.