Other mistake: When the AT-ST (Imperial Walker) blasts the shield generator, when it explodes, you can see the glowing remains of it (only visible on widescreen DVD). Note that it was fixed on the 2019 4K release.
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. (01:26:40)
Revealing mistake: When Captain Needa is killed by Darth Vader because they lost track of the Millennium Falcon, in the background, two guards come to take him away. You can clearly see the dead captain get up almost by himself. (01:10:50)
Other mistake: When in the cave, Leia falls and is caught by Han. She tells him that "being held by you isn't quite enough to get me excited." As she is saying this you can see Harrison Ford mouthing her exact words. (00:44:05)
Audio problem: After Luke frees himself from the cave in the beginning, he stumbles towards the exit. You can hear his lightsaber go off, but it stays firmly on. Cut to him emerging, and now his lightsaber's off. (00:09:30)

Continuity mistake: When one of the stormtroopers unties Han Solo before he is put into the carbonite, watch his wardrobe - he has a disappearing / reappearing jacket on. This is fixed to an extent on the DVD - the jacket is now lightened to almost match his shirt, but it's still shiny like leather. (01:32:05)
Suggested correction: It just looks like a jacket because it's a shadow.
Revealing mistake: When Leia and the crew return to cloud city to rescue Luke from the scaffolding, watch very closely as Lando ascends up the lift, just after the top hatch opens from the overhead angle, you will see that Lando has been replaced with a G.I. Joe look-alike figurine. (01:55:10)

Continuity mistake: After Luke says to Yoda, "You want the impossible," Yoda's position jumps from the ground to a rocky boulder, where he then lifts Luke's X-Wing fighter from the swamp. (01:11:30 - 01:13:05)

Visible crew/equipment: This can only be seen in the widescreen version. When Han, Leia and Chewie venture out of the Falcon to investigate the cave they're hiding in on the asteroid, as they walk towards the left of the screen you can see the edge of the tarpaulin sheet they're standing on as well as the faint outline of a crew member trying to get out of the camera shot. (00:58:40)
Continuity mistake: After Boba Fett's ship, Slave 1, takes off, there's a close-up of Leia's face. In the background, Lando can be seen standing erect, putting his hands on his waist. In the next shot, he's crouching, with his hands in front of him. (01:39:25)
Revealing mistake: During the attack on the rebel base, after the headless AT-AT falls on the ground, another one shoots at the rebels. You can see the pyrotechnic charges where the impacts are. (00:33:05)
Continuity mistake: When Han, Leia and Lando are walking down the corridor to the room with 'refreshments' (aka Darth Vader), Han and Lando keep switching sides. (01:27:55)
Continuity mistake: Near the end of the film, when the Millennium Falcon is trying to get away from Vader, there's the scene in which Vader asks the admiral if he disabled the hyperdrive on the Falcon. This scene is backwards, as you can tell by the insignia on the admiral's chest. Normally it should be on the left side of the chest, but in this scene, it is on the right side of his chest. This was fixed on the DVD re-release. (01:57:10)
Continuity mistake: When the AT-AT that Luke destroyed with a thermal detonator falls over, Luke is nowhere to be seen, even though he should be visible, judging from his position in the previous shot. (As noted in another mistake, the AT-AT is suddenly further away from him after he falls to the ground, but he is still to close to be out of shot when it falls over.). (00:31:45)
Continuity mistake: When Leia and Lando are holding off the stormtroopers while Chewbacca and the droids escape into the Millennium Falcon, a stormtrooper comes running through the doorway. You can see the burn mark on his armour before Lando shoots him. When the trooper is shot, another trooper comes running through the door, also with a burn mark on his armour. Then Leia shoots another (or perhaps the same) trooper, and the burn mark is once again there before the trooper is hit. (01:44:20)
Revealing mistake: When Luke and Leia are watching the Millennium Falcon leave at the end of the movie, its light goes through Luke's arm. (01:56:25)
Continuity mistake: After Luke crashes during the battle on Hoth , the AT-AT is far away from him and is moving rather slowly. A few shots later, the AT-AT stomps on his snowspeeder, but it was too far away from him in the previous shots to reach it that quickly. (00:29:55)
Continuity mistake: At the end of the movie when the Millennium Falcon is escaping from the TIE fighters we see some going after the Falcon then we see a shot of Vader looking at it - there are now no TIE fighters. They had no time to escape away. (01:57:45)
Revealing mistake: Right before Luke nicks Vader in the shoulder with his lightsaber, Vader cuts through a railing. There's a hole in the railing exactly where Vader apparently cuts it. Also Vader's lightsaber never comes into contact with the railing, yet it sparks. (01:50:15)
Revealing mistake: When Luke and Vader are fighting in Cloud City, there is something looking like tape wherever they hit something with their lightsabers. Maybe it's pyrotechnic charges?