While Kirk is talking to Pike in a bar he has a nearly empty drink. Then during a shot when he lifts the glass to take a sip, the glass is full. In the next shot, when he actually takes the sip, the glass is nearly empty again.
When the Enterprise's trash exhaust port is first shown, it's about half-way up the secondary hull and has sliding panels on either side of it. Almost two minutes later, the panels are missing and the exhaust port appears to be farther down and forward.
Admiral Marcus is holding Khan's frozen comrades hostage as leverage to force Khan to work for him, yet somehow Khan was able to get access to his people, and conceal them into the 72 photon torpedoes he helped construct. No one witnessed this, and no one noticed that Khan's people had vanished.
After the final battle with the dreadnought class ship, when the Enterprise's engines fail, it starts dropping towards Earth. However, during the Battle both ships and the debris from the battle are clearly in a stable orbit which does not require power to maintain.
If the Vengeance and Enterprise were both traveling at Warp, the Vengeance shouldn't have been able to fire on them with phasers. Photon torpedoes are stated to travel faster than light but not phaser beams.
The goof items below may give away important plot points.
During the exchange negotiations between Khan and Spock, Khan says he will destroy the Enterprise's life support, located behind the aft nacelle. The nacelles are the two tube-like structures behind the saucer section, one to the right and the other to the left. There is no 'aft' nacelle. Also considering the nacelles protrude out behind the ship, there is nothing behind them.
When the Starship Vengeance is approaching San Francisco (when you see San Francisco in the background), the sun is in front of the ship, however when the ship hits the water after destroying Alcatraz, the sun is now behind it.
When Kirk grabs Scotty from the walkway, there is clearly no one nearby, and gravity is clearly sideways from the railing. So there would be no way for Chekov to appear in time to grab Kirk's hand, even if he had been working his way along the railing.
On board the USS Vengeance, Scotty successfully stuns Khan for several moments with a single shot from his hand phaser. Later in the movie, when Spock and Uhura are battling Khan on Earth, Uhura shoots Khan with multiple direct hits from her hand phaser with no effect.
When Spock is chasing Khan in San Francisco he is holding a phaser in right hand until the scene where is jumping on to the transport where he has empty hands, until he climbs to the top of the transport where he again has the weapon in his hand so that Khan can kick it out.