Continuity mistake: When Indy and the German driver crash through the villager's construction ladders the truck is missing the Mercedes-Benz hood ornament that appears in every other shot.
Audio problem: When the monkey is crawling on Marion's head at Salah's place, the sound does not match Marion's mouth.
Continuity mistake: In the scene where Indy is telling about the Staff of Ra, the background color of the blackboard changes between shots. It looks like there are some chalk residues (after the blackboard has been washed) in one of the shots, but not in the other.
Factual error: The German submarine that Indy stows away on has the number 26 on its side. The U-26 was a Type IA U-boat (it came into service in 1936 and was sunk in 1940), but the sub in the movie seems to be a Type VIIB or VIIC. It's true that the two types looked fairly similar, but they were different classes of U-boat.
Continuity mistake: In the scene where Sallah and Jones are in the Well of Souls where they are putting the sticks in the Ark to left it up. Notice Sallah stick, when he first puts it in, on the stick are no tassels. But when they both left at the same time Sallah same stick that he put in has tassels on it.

Revealing mistake: Just before Belloq says "beautiful", look carefully at the woman's ghost in the center of the screen. You can see one of the wires on her back that makes the stunt woman fly, a black line above her left shoulder.
Continuity mistake: The fight scene in Marion's bar clearly shows Indy using an automatic pistol to shoot at the bad guys. You can see the slide lock open when he runs out of ammo. But when the scene switches he is holding a revolver. Then when the huge guy is strangling Indy and the German says, "Shoot them, shoot them both." Indy is once again holding an automatic.
Audio problem: Watch and listen when Marion is being held prisoner in Nepal by one of the thugs. As he seizes her, she's grunting in strain/pain. We can see this in the first, frontal shot of them. However, in the following side shot (from slightly behind Marion as Toht walks over with the heated poker), Marion's mouth is completely still and almost shut, yet her sounds continue (and these sounds would have to be made with wide-spread lips).
Factual error: The captain of the Bantu Wind, Katanga, wears a hat that originated from the Yugoslav People's Navy. But the Yugoslav Navy wasn't formed until 1945, 9 years after this movie takes place.
Factual error: During the fight scene at the German flying wing, the pilot ducks into the plane and pulls out a pistol and cocks it. The pistol is a P38 9mm semi-auto that the Germans did not issue until 1938, two years later.
Continuity mistake: As Indiana writes "Neolithic" on the blackboard in his classroom at the beginning of the movie, he stops and checks to see if he's spelling it right, then finishes writing it. Later shots of the board reveal the same word on the board, only either the handwriting is different or the actual structure of the word has changed.
Continuity mistake: In the map room, Indy pulls out a notebook and takes off the elastic band. The elastic goes flying away. Moments later he closes the book and the elastic is in his hand. Also, the elastic starts off being pink (the one that flies off), then blue (the one that he puts back onto the book) and ends as tan coloured when he puts the book back in his pocket.

Revealing mistake: After Marion goes in the basket, there is a chase and a truck tips over, if you look amidst the dust, you can see the ram used to put the vehicle on its side.
Plot hole: Indy stows away on the Nazi sub on the way to the secret Nazi base somewhere near Greece. Where did he stay all that time? Inside? On a VII-Class boat, there's barely enough room for the regular crew, and hardly any places to hide. Also, everybody knows everybody, so no passing yourself off as "the new guy." Outside then maybe? Well, the boat did dive, it's in the dialog, and while on the surface, German U-boats always had a bridge watch on the tower, so no luck there either. Note that by all accounts I could find, German submarines were required to always post a bridge watch, if only because otherwise they would be running essentially blind. It's not like the helmsman's station in the tower had any windows.
Factual error: When Marion is thrown into the pit of snakes and grabs the statue to prevent her fall, underneath her skirt she is wearing black spandex shorts. Shorts such as these did not exist in the 1930s time period of the film.
Factual error: The truck that turns over and blows up after Indy shoots the driver is a Mercedes Benz L701. It began production in 1944, and this movie is set in 1936, so the truck is not period correct. Also, the alleys are too narrow to allow this truck to turn a corner.

Continuity mistake: The position of the Top Secret sign and the whole text is put on the ark's box across vertically laid planks, with a wood crossbar on the right, which only the side panels have. A shot later the sign appears slanted across horizontally laid planks. Though it could be argued that there could be a second sign on the opposite side with horizontal planks, when the warehouse guy turns left to store the crate, we get a glimpse of the left side and it's not painted.
Suggested correction: The other text and the different plank design could be on the base of the box or on the front wall (the wall opposite the man pushing the box), two walls we never see when the box is being pushed. In fact, at one point the camera passes next to a box that has two texts, one on a short wall and one on a long wall.
Factual error: The interior of the German U-boat is completely different from the real interior of a Type VII U-boat, which is correctly shown in movies like 'Das Boot' or 'U-571'. For example, the ladder from the conning tower faces a different direction, the map table is too big and on the wrong side, the valves for submerging are on the side of the control room and not at its front and stern, depth gauges are too big...etc. All interiors were standardized, so it could not have been an individual build.
Continuity mistake: After Satipo runs with the idol, Indy grabs a root to help him climb up the ledge. Half a second later, from a different angle, the root is meters away and he is grabbing it again, repeating the whole scene.
Continuity mistake: Just before the car carrying Marion in the wicker basket explodes in the alley, Indiana shoots a man twice in the chest. The man has blood stains on his shirt before he is shot.