Continuity mistake: Admiral Marcus' ship "The Vengeance" comes out of warp and immediately appears to be right beside the Enterprise. The top of the ship even goes over the roof of the Enterprise to prove it. Yet from an immediate side angle of the two ships, the distance between the two ships has increased noticeably and the Enterprise is now facing towards the Vengeance.
Factual error: At one point Khan threatens to target the Enterprise's life support systems which are located behind her "aft nacelle". The Enterprise doesn't have an aft nacelle; as the many exterior shots show, it has a port nacelle and a starboard nacelle. Khan a) is familiar enough with Starfleet ships to know this basic fact about the Enterprise, and b) can see its nacelles for himself on his own viewscreen while he's delivering this line. There is no "nacelle" housing the impulse engines. They are enclosed by the hull of the saucer section.
Continuity mistake: After the Vengeance attacks the Enterprise and Scotty temporarily disables the Vengeance, Kirk and Spock head toward Sickbay. They pause in the corridor and have a brief exchange about "logic" and "gut feelings," with the camera cutting back and forth between Kirk and Spock. When the camera is on Kirk at first, the corridor behind him on his left is clear; when it cuts to Spock, a girl in a red engineering uniform with her hair in a bun and a glowing handheld device appears on Kirk's left and walks past, seemingly without noticing them. Camera cuts back to Kirk and the corridor is clear again. When the camera cuts back to Spock, the exact same girl with the hair bun and handheld device appears again on Kirk's left and walks past again, in the same direction as the first time, but this time looking over her right shoulder and into the camera. Camera quickly cuts back to Kirk and then to Spock, and now the corridor behind Spock is completely clear - the girl has vanished entirely.
Other mistake: After the Enterprise is disabled by the Vengeance and falls out of warp, Kirk and Spock have a discussion in one of the corridors about letting Khan help incapacitate the attacking ship. Watch behind them. The ship is running under Red, but the crew members are walking past them calmly, some even watching their conversation as they pass! Not really Red Alert protocol... In fact, through all of this, only one crewman runs past in the entire sequence.
Other mistake: As Kirk realigns the warp core to restore power and save the USS Enterprise from crashing to Earth, Sulu tells Spock: "Without power, we'll incinerate on re-entry." It shows Kirk restoring power after it plummets below the clouds, which is well beyond the point where they would have been incinerated.
Plot hole: During the warp-speed chase, the Vengeance literally blasts the Enterprise to pieces, and dozens if not scores of Enterprise crew members are killed and injured in the carnage. The medical crew, including Chief Medical Officer McCoy, should have been working feverishly on the wounded and dying for hours, at least. Instead, as Kirk asks Khan for help, the Sickbay is practically deserted, and McCoy is almost idly conducting blood experiments on a dead tribble. There's no sense of a catastrophic medical emergency whatsoever. It's as though the Sickbay sequence was shot for a different script in which there was no emergency, and then lazily inserted into a rewritten script.
Continuity mistake: As Kirk and Spock converge in the lobby of the Daystrom building, Spock greets Kirk with "Captain" and Kirk responds "Not anymore Spock, First Officer." There is a cut between "Not anymore" and "Spock." During the cut Spock moves from being in front of Kirk to behind him and the two have moved from the middle of the lobby, with the ramps visible 10 to 15 yards ahead of them to right beside the ramps.
Factual error: During the Vengeance's fall to Earth, just after hitting Alcatraz, the saucer section hits the surface of San Francisco Bay, the outer rim diving dozens of meters into the water and its rear lifting up almost vertically into the air. A few seconds later the ship is seen from the onshore view, and it is straight and level and heading directly at Star Fleet Headquarters. Hitting the water at that angle and speed, so close to the shore, the ship could not have righted itself and gotten on that trajectory going that fast. It should have sunk or flipped over when it hit the water. Even if it could flip 360°, there was not enough time, distance, or height for the ship to continue on to its target at that speed. The water's drag would have slowed it too much.
Continuity mistake: After Harrison (Khan) has attacked the senior officers' meeting and Pike has died, the position of Pike's head on the cushion changes between shots. When Spock disconnects from the mind meld, the stitching on the cushion is above Pike's ears. When Kirk checks for a pulse, the stitching is below Pike's ears. The angle of his head also changes.
Continuity mistake: In the bar, Pike meets up with Kirk to tell him that they gave him the Enterprise. When Pike tells Kirk that Spock has been reassigned to the Bradbury, Kirk pours liquor into his glass, nearly covering the big round ice cube. When Kirk lifts the glass from the bar, half of the liquor instantly disappears.