When Penny is talking to Will about jettisoning his body into space, the package in her hand suddenly disappears when she claps her hands together. Immediately afterwards, she retrieves it from her left.
When Dr Smith is talking to the Global Sedition Man in the Holo-Desert. He screams "Where is my money". The next shot is a close-up of his face and he has his mouth wide open, yet he still manages to say nearly a whole sentence without moving his mouth.
When the spiders are about to break through the door on the Proteus, a shot from the robot's point-of-view shows John Robinson saying "Will, we have to get that door open," a few seconds before his mouth moves.
After Major West falls over startled from the spiders behind him, you see him take aim to shoot the spider on his chest. In the next shot, you see the flare from the gun firing, but there's no sound. Two shots later he fires the gun, and it functions properly.
During the attempted destruction of Jupiter, there's a brief moment (right after Major West gets blown back by the electric blast) where you hear the robot's pulse blaster firing, but don't see a beam.
During the escape scene on the Proteus, Major West converts his pistol into a rifle with a series of add-ons. However, right before they boarded the Proteus (and Will was demonstrating how he had taken control of the robot), a rifle is visible in the armory that John gets a pistol from. Why wouldn't West take a rifle in the first place?
When the robot is removing the restraining bolt, a close-up shot shows him using his left arm. The long shot of the robot removing the bolt shows all his arms (including the left one) waving about in the air.
After Penny opens the package containing the ladder she escapes with, she stands up. The package is in her left hand by her knees, and her right hand is down at her side. In the next shot, she's cradling the package in her right hand up at her waist.
In the opening battle scene, Jeb's HUD displays a "System Overload" warning after he was shot by the Sedition raider. That warning was actually displaying since the moment he launched. It also disappears right after Don moves in to rescue him.
As the Jupiter I is lifting off from Houston, the pyrotechnics used to simulate the rocket motor blast actually causes the entire top half of the ship model to jump up and separate from the lower half briefly.
Right after the time bubble collapses (near the end of the movie, on the alien world), the Jupiter launches for the first time (without the professor). Since the bubble had collapsed, they (professor and aged Will) shouldn't have been able to see the escaping Jupiter: the doorway to the past had closed.
In the (repeated) end sequence when the Jupiter 2 is trying to take off from the planet, it should not have had to dodge the same fatal rock the second time, as there was a significant takeoff delay when John Robinson returned.