The explosion of a failed Extremis subject is so strong it vaporizes living organisms in the blast radius along with collateral damage (as shown at the Chinese Theater). In Tennessee, the site of a similar incident, it is shown that the crater of the explosion is a few feet from the corner of two walls. The walls, for some reason, were not obliterated even though the explosion was strong enough to destroy everything else.

The movie takes place in the year 2259. Every reference to Khan's hibernation stated that he was frozen for "300 years", which would mean that genetically engineered superhumans were created in the year 1959. Naturally, there's some leeway when someone refers to "300 years" but even setting the threshold to 250 results in the year 2009. It would have been more convincing if they had set it at "200" or "over 200 years".

During Bond's fight with Patrice in the Shanghai high-rise, Patrice's sniper rifle fires several times, presumably by accident. However, moments earlier, when Patrice assassinates his target in the neighboring building, the rifle is shown to be bolt-action. A bolt-action rifle can't be fired more than once without manually loading the next round.

Dr Schultz pays with 12 $1000 bills, but $1000 bills were not created until 1861, after the time period of this movie.

The wound on Jack's nose changes throughout the film.

After Gatsby tells the story of how he met the sailor there's a blue shirt on the bed which disappears when he exits the library.

When Pat runs towards the school, he talks with Mrs. Metgers and in the scene where they're walking up the stairs you can see a crew member with a white cap reflected in the glass of the door.

Grug's loin cloth gets ripped by a lion at one point. The rip magically fixes itself just moments later.

Iranian revolutionary guards didn't have caps with their uniforms until 1988 (after the end of the Iran-Iraq war).

When Jack reverses aggressively out of the motel carpark onto the road, he steers the car to his left and drives down the road. Note that there are no visible tyre marks left by Jack's car as he drives away. In the next shot, the police cars exit the carpark sliding out onto the road to their right, following tyre marks which are already on the road. The tyre marks on the road are likely from a prior take.