Revealing mistake: After the Nazi tank drives off of the cliff and hits the bottom, the upper turret breaks off. As the turret rolls over, you can see there is no hole in the bottom, just in the top.
Continuity mistake: When Indy meets Hitler, there is a shot over Indy's shoulder that shows his lips are slightly parted (relaxed). Cut to Indy's face and he closes his mouth tightly. In the very next shot, with no elapsed time, we see from Indy's shoulder again that Indy's mouth is relaxed as before. (01:10:15)
Other mistake: While Indy and Henry are aboard the zeppelin Indy tosses Vogel out the lounge window, and when Vogel shouts up to the departing airship, in the next exterior shot we see the zeppelin airborne against a scenic view. Later, when Indy realises the zeppelin is turning around it cuts to another exterior shot of the airship, and the scenic view is identical to the earlier shot, only it's been flipped. The images of the zeppelin facing two different directions were composited onto the same scenic background.
Factual error: When Indy and Marcus first arrive in Venice on the boat the dock they alight on to has tactile pavers (for the visually impaired) installed on the platform. These weren't invented until 1965, some 25+ years after the scene takes place.
Continuity mistake: Onboard the ship at the end of the prologue, after Indy grabs the cross before it falls over the side, the guy in the white suit climbing up the steps. He's only reaching the top as the shot changes. In the next shot, he's already a few steps away from the top of the steps, telling his men to stop Indy.
Continuity mistake: During the temple quake sequence towards the end of the film, Elsa fails to maintain her grip on the rising earth and falls onto tilted ground, sliding towards the chasm. At this point, Sallah is kneeling on the ground, but in the very next shot, he is now grabbing hold of Henry.
Continuity mistake: When the Hatay soldier is decapitated in an attempt to navigate the temple's defences, his head rolls along the ground and Elsa turns her head around and looks over her shoulder. In the next shot, she has now turned her full body around, and in the shot after that, she is looking over her shoulder again.
Continuity mistake: When Sallah rides up to the tank to rescue Henry we see Indy toss the whip over the side of the tank, but after the tank falls over the cliff, when Indy climbs back up then walks over to stand between Henry and Sallah the whip is coiled and back in its place, hanging from his belt. Even Indy can't do that. (01:36:00 - 01:37:40)