Audio problem: When Mary is reading Peter Pan to Gertie, her shadow's lips don't match up with the words she is saying.
Continuity mistake: In the final scene of E.T., where E.T. is saying goodbye to Elliot, E.T. rubs Elliot's shoulders, then removes his hands from Elliot's shoulders. The shot then changes angles from behind Elliot to behind E.T. Upon changing the angle, E.T.'s hands can be seen back on Elliot's shoulder. When the shot alternates between the two angles (for about ten seconds) so does E.T.'s hands alternate being on or off of Elliot's shoulders.
Deliberate mistake: At the end of the film when E.T. is leaving, there is a bright light from the ship, and wind and dust from the spaceship's liftoff. But when the spaceship has lifted off and has flown into space and made a rainbow, they show Elliot and the others and there is still the shining light, plus wind and dust blowing for no reason.
Factual error: The government operatives go to great lengths to cordon-off and sanitize Elliott's home with extensive hazardous material preparation. Controlled airflow, Hazmat suits and everything. But as soon as E.T. "dies," everybody takes off their Hazmat masks. So, E.T. was only a Hazmat threat when he was alive? A real government Hazmat team would never drop their guard like this, as long as the alien body was still present.
Visible crew/equipment: When Elliot is luring ET with Reeces Pieces and goes into his room and through his drawers to find more candy, you can see the shadow of the crew moving on the wall next to Elliot.
Continuity mistake: When E.T. is in the kitchen reading comics, he glances over at the TV then takes a look at the coffee table which has, among other items scattered on top, six pack rings. A few shots later E.T. is at the coffee table working to create a communication device, but now there's one can of beer attached to the rings.
Continuity mistake: When Mary gets her first look at an ailing E.T. lying on the bathroom floor, she panics then hurries everyone downstairs - leaving E.T. behind. Moments later one of the astronauts happens upon E.T. and now a bright pink bath towel can be seen draped over the bathtub's edge, which wasn't even there to begin with.