Continuity mistake: The microphone on Red Leader's helmet switches from one side of his face to the other, then back again as he pulls out of the Death Star trench (just before he crashes). (Actually the entire shot is mirrored. Check out the scuff marks on the centre of his helmet, they change direction as well.) Other shots and pilots' mics switch too: Red Leader when he says, "All wings, report in", Wedge after a shot of Leia and C-3PO monitoring the battle, Luke when Red Leader says, "There's a lot of fire coming from the right side of that deflection tower" and when Luke says, "I'm hit, but not bad".
Continuity mistake: When Luke, Han and Chewbacca attack the prison of the Death Star, in one shot the end of the corridor is in a slope while it is supposed to be horizontal. (It looks to me like it's actually a flat panel painted to look like the corridor. The angle looks perfect for the opening shot of them entering the prison floor, it just looks "off" for any other angle.)
Continuity mistake: Right before Luke yells at Han after he fires his blaster in the trash compactor, there's a wideshot where Han is sitting among some garbage and holding his blaster with one hand. In the next shot, he has turned the other way and is standing, holding the blaster with two hands. (01:16:15)
Deliberate mistake: While Luke hears Ben's voice telling him to let go, Vader says, "The Force is strong with this one," and in the following wide shot as they say Luke's computer is turned off, the dent on C-3PO's head is on his right side instead of his left, revealing a flipped shot. (01:55:50)
Deliberate mistake: During the scenes of the Star Destroyer attacking Tantive IV, and also when the stormtroopers board the ship, the Alderaanian consular security force are wearing helmets with an attached communications device at their left, but in a few flipped shots the devices are at their right side of the helmets. (00:02:35)
Continuity mistake: In the beginning of the movie, aboard Blockade runner, a big smudge of dirt appears on top of R2-D2, close to his square-framed lens 'eye'. This was not present a minute earlier and when the droids have landed on the desert, the smudge is gone again. (00:02:40 - 00:09:50)
Continuity mistake: During the medal ceremony, there is a close-up of Han followed by a shot of Leia with a big smile. She turns to the right to get a medal, then there is a wider shot of her and she's not smiling.
Continuity mistake: When the walls of the trash compactor start moving, Han is holding the pole, and leans it against the wall to his left (we hear it hit the wall). In the next shot, the pole leaning on the wall to his right instead., and the right side of the pole moves up and down between the next few shots.
Continuity mistake: Widescreen version: In the trash compactor, before the scene where the droids leave the control room, both ends of the pole used to stop the compactor are coming into contact with the walls. After the droids leave the room, the right end of the pole is not coming into contact with the wall. (01:18:50 - 01:19:45)