Continuity mistake: When Indy is rescuing his father from Brunwald castle, he uses his bullwhip to smash through the window of the room his father is kept in...but in the time it took to jump into the room and get smashed over the head by his father, the bullwhip he used would have lost lots of its momentum and it wouldn't be waiting for him to grab it and use it later on the movie. (00:47:40)
Continuity mistake: The back of the boat gets chewed up by the prop in the same place twice.
Continuity mistake: When the driver of the tank is shot and he changes the direction of where the tank is going, Indiana gets up, with his strap still stuck on the nozzle. However, in the next shot, he's become clear of being stuck. There's no way he could have gotten his strap undone so quickly.
Continuity mistake: Near the end, in the "Leap of Faith" scene, Jones steps off the edge of the abyss and discovers an ingeniously-camouflaged stone bridge. The camera pans to show the camouflage effect from another angle, and it's obvious that the stone bridge is only about one foot thick from one end to the other. Relieved, Jones scampers across to the other side, and the camera cuts again to show that the bridge is now a full three feet thick.
Continuity mistake: Young Indy escapes from the four diggers, and you can see them exit the cave after him as he pauses. He moves down to and jumps onto the horse (missing the first time). He gets on the horse and you see the diggers, starting running from the exact same place that they were. It's like they paused their pursuit while the camera was not on them. If they kept running, they would've easily caught up with him before he jumped down.
Continuity mistake: When Indy sees Elsa losing her grip on the tilted earth and then falling on the rising ground during the temple quake sequence towards the end of the film, he lets go of his father, but when the camera changes to behind Elsa in the next shot, Indy is back holding his father again.