As McCoy regains consciousness in Zarabeth's cave, his hair changes from frosted to normal in one cut.
When Kirk walks into the time portal from the medieval past, he doesn't have his communicator on his belt. But as he appears back in the future, the communicator is prominently on his right hip.
When Kirk is walking and interacting in his timeline, there is no question or reaction to his uniform or appearance which are out of place in the time period.
When Spock, Jim and Bones meet the real Mr. Atoz he is sitting on a bench without any cushion. A few seconds later, when he offers Jim to sit on the bench, suddenly there is a cushion on the bench.
If planet Vulcan really were millions of light years away, it would be well outside the Milky Way and traveling to the nearest galaxy from there would take at least 300 years at maximum warp.
When Bones and Spock return through the portal, they leave the past wearing fur coats - they arrive in The Library without the coats. The coats are from the ice age, and like a person who had been "prepared" by the atavachron (the time portal machine), they cannot survive traveling forward in time.
When in the Ice Age period, Spock starts regressing at the level of his ancestors five thousand years earlier. However, McCoy does not. Behaviorally speaking, humans have remained relatively unchanged over the past 5,000 years, despite our technological advances. The Vulcan people were on the brink of oblivion not because of what they could do technologically, but who they were emotionally. Humans now are similar to humans at the beginning of the Bronze Age, regardless of our technological advances.
When Bones and Spock are outdoors in the ice age, it's very cold, below freezing. We know this because Bones later complains of frostbite and also about the abundance of snow. Yet we can't see their breath.
Even without preparation of the person, it is inconceivable that the Atavachron would allow a modern weapon to be functional in the past. A weapon like a phaser would allow major devastations in a less sophisticated timeframe (see The Omega Glory for example).
Zarabeth's Ice Age cave has a smooth, perfectly flat reinforced concrete floor like those found in a TV studio.
When Spock and Zarabeth are talking, Spock turns to see McCoy standing at the cave entrance and McCoy says, "You've been dishonest with me, Spock," but his lips don't move.
The City on the Edge of Forever (1967) established that the presence of one man can change the future; Tomorrow Is Yesterday (1967) established that the absence of one man can do the same. While the exact or even approximate population of Sarpeidon is never stated, there was obviously far more than one person sent back into the past, and each one had the potential to contaminate the planet's timeline. No character ever raises this problem on screen.
The fact that phasers had not been invented in a certain time period would not affect the physics of how phasers work. If that idea held true, a metal implement of any kind, such as a knife, would no longer function if transported back to a pre-metalworking age. However, the Atavachron could have selectively inactivated weapons.
When the prosecutor is talking to Kirk, the mort (female thief) in the next cell accuses Kirk of being a witch. The constable confirms this, saying that Kirk talked to unseen spirits, one of which he called "Bones." Although Kirk did address Dr. McCoy as "Bones" through the unseen time portal, the constable was not present when he did so.
When Dr. McCoy is confronting Spock and Zarabeth about whether or not he and Spock can leave the Ice Age, Spock gets enraged at McCoy for accusing Zarabeth of lying to them. Spock gets up and throws McCoy against the rocks behind him. As he is carrying McCoy towards the rocks, the shadow of the boom mike following them can be seen on the higher rocks in the background.
When Spock tries to use his phaser to warm a boulder at the base of the ice cliff, it doesn't work - presumably because phasers didn't exist in that time period. But when he lays McCoy out in Zarabeth's cave and examines him, the doctor's medical tricorder seems to work just fine.
Zarabeth tells Spock "There aren't many luxuries around here", yet she has perfectly shampooed and coiffured hair. And she is also wearing facial makeup, which would hardly have existed at that time.
In the ice age Bones tells Spock "My hands and face are frostbitten. I can't feel my feet." He should know, he's a doctor. Yet after he warms up in the cave he's perfectly fine again and has suffered no ill-effects from his frostbite and it's never mentioned again.
Spock tells Zarabeth that he comes from a place "millions" of light-years away. The Milky Way is approximately 100,000 light-years in diameter - so "millions" of light-years would place Vulcan well outside the galaxy. This may be a sign of the dementia he suffers as a side-effect of time travel.