Revealing mistake: While Poe leads the aerial assault on the oscillator, in the closeups of Poe and other X-wing fighter pilots, note the cockpit windows at the pilots' left and right have identical marks and scratches in their individual X-wings. It's particularly noticeable when one closeup follows another.
Visible crew/equipment: When C-3PO is on the conveyor belt, if you look in the reflection in his head you can see the camera crew.
Revealing mistake: When Amidala and Anakin are eating and he cuts her a piece of the fruit and "floats" it back to her, the bite appears in the fruit a split second before she actually eats it. [This appears to be fixed in rereleased versions]. (00:53:40)
Continuity mistake: Darth Maul's neck is black, but when he is talking to Darth Sidious on Coruscant, his neck is flesh-toned. (00:38:45)
Continuity mistake: When Lando says "Go on, you pirate" his entire outfit is reversed - his shoulder holster strap goes the other way, and his rank insignia changes side too. (00:50:00)
Continuity mistake: After the speeder chase, everyone lands in quicksand. In one shot, 3PO is buried headfirst. In the next, he has flipped and his upper body is visible.
Other mistake: Towards the end, Chewbacca and Beckett have walked some distance from the landed yacht. They come up a small slope and are faced with Han, waiting for them. But an aerial shot shows that slope is very short and just runs down to a cliff. There's no way they could have got to the base of the slope to walk up it in the first place.