Revealing mistake: While Indy and Marion are in the Well of Souls, and they encounter the snakes, Indy falls to the ground only to be confronted by a hissing Cobra rearing its head. Look carefully and you'll see the snake's reflection on the safety glass between it and Indy. Briefly you can also see the torch's reflection while he's waving it around. [This has been corrected in the new DVD set. You can only see it if you have the original VHS. I think it's visible in the "making of" on the DVD as well.]
Continuity mistake: When Indy goes to save Marion in the tent he removes the handkerchief gag. But when he decides to leave her there, he puts the gag back on and some of her hair gets caught underneath the gag on one close up shot from the left. But when the camera goes to the back of Indy, her hair is no longer caught underneath the gag.
Continuity mistake: Near the beginning of the movie where Indiana is running away from the tribe of South American Indians after Belloq robs Indiana of the gold statue, in one shot you can see that the Indians are only a few meters behind Indiana, but in the next shot, they are at least 50 meters behind.
Revealing mistake: Watch the shadow of the stone Indy pushes out to escape from the Well Of Souls - it bounces down the sand dune, exactly the way a large rock shouldn't.
Continuity mistake: When the village crowd separates and reveals a standoff between Indy and the black-robed swordsman, the rear shot of Indy shows his hip-bag to the rear of him, the frontal shot of Indy shows his hip-bag on the forward part of his hip.
Revealing mistake: After Indy gets behind the wheel of the truck containing the ark, he jerks the steering wheel to the right to run the pursuing jeep off the road. He then jerks the wheel to the left to hit an upcoming motorcycle. In neither case do you see his body sway with the presumably drastic swerve of the truck. Apparently this was filmed in a stationary vehicle, or one in tow that was fitted with a prop steering wheel.
Revealing mistake: When Indy jumps across the pit to pursue the spy that took the idol, he barely makes it across, grabbing the ledge on the opposite side with his hands. When he pulls himself up, he kicks loose a rock and the entire "solid" rock wall shakes like it's nothing more than a foam wall.
Continuity mistake: In the Himalayan bar scene when the sherpas leave, Marion starts to pick up the shot glasses and puts them on a tray - there's a large group left at one end of the table, one in the middle, and two by a bowl at the other end. When she returns for the rest, the two by the bowl have vanished.
Continuity mistake: When Belloq is talking to Indy and Marion in the Well of Souls, he's shown holding his hat with his right hand. But in the only over-the-shoulder shot (looking into the Well of Souls) the hat is laying on the ground and Belloq tosses a stone down with his right hand. Then the hat is back.
Revealing mistake: When Indy fights the driver of the Nazi truck, he gets thrown through the windshield, and eventually ends up underneath the truck. When he is climbing underneath the truck, heading towards the back of it, you can see the trench that was dug down the middle of the road to provide the stuntman (this being one of the few stunts that Ford didn't do himself) with the extra room needed to do this feat. (This is backed up by DVD commentary. The director stated that the truck was too low to the ground so they had to dig the trench in order for there to be enough room for the stuntman).
Factual error: When Indiana Jones is fighting the big German by the airplane, the German punches Indiana Jones on his right cheek. Indy's face and body are both knocked down to the right, in the direction of the punch. When you watch this, you can see it's impossible to be knocked down in such a way.
Factual error: The film starts in the year 1936, and just like the RPG Indy uses to threaten Belloq with, the MP40s (or at the earliest, MP38s) are an arms-historical anachronism in this movie. As the numbers in their names imply, the MP38 was first produced in 1938. The weapons in the movie are not the similar-looking ERMA MP-36, you can see the ribbed, vertically angled magazine holders typical for the MP40 - its predecessors had either a 30°-canted, smooth magazine holder (MP36) or a round viewing port for checking the ammo (MP38).
Continuity mistake: At the beginning of the film, Indiana Jones uses his whip to disarm a man with a handgun. The scene showing this action includes several quick shots, but as a result of the use of these multiple shots it can be seen that the whip descends twice. (Slow motion helps, but isn't required.)
Factual error: Indy's satchel is a Mk VII gas mask bag, which was created for use in World War II, so it didn't even exist at the time the film takes place (1936). The similar Mk V existed at the time, though.