Continuity mistake: When Indy breaks through the window of the room where his father is locked up, you can see that it is cloudy and raining outside. However, after Henry says, "Junior, you did it!", in the next shot, there is a very intense and constant brightness (not the instantaneous brightness of lightning) coming through the broken window, as if it were sunny. This brightness disappears two shots later. (00:50:00)
Continuity mistake: When Indy climbs onto the circus train, he begins to be chased by Half Breed and Fedora. However, in the next shot, when Indy enters the reptile car, it is now Rough Rider who is behind Half Breed. (00:06:22)
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Visible crew/equipment: After Indy and Henry have escaped from Castle Brunwald, Indy jumps into one of the boats, pulls the motor starter cord and jumps back out, then just as he bends over to release the boat from the piling, right between Indy's legs the black covered arm of a hidden crew member appears from under the tarp taking hold of the throttle, steering the boat away from the pier. (01:02:40)
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Visible crew/equipment: When Young Indy is trapped in the train car with the lion, he tosses up the whip to Fedora, and as he's being pulled up in the interior shot, he's wearing a stunt apparatus around his waist above his beaded belt. [The stunt double is also wearing a glove on his right hand holding what looks like skewered meat, to presumably draw the trained lion, but that's a lot harder to see without pausing]. (00:08:35)
Revealing mistake: In the catacombs of the library, Indy and Elsa are waist deep in petroleum. Indy has a torch, and if you look carefully, you will see burning pieces of the torch fall and hit the petroleum. Wouldn't this start a fire as Kazim later on sets the cavern alight with a single match? (00:34:05)
Continuity mistake: In the library scene Indy discovers the "X" high up on the balcony. The X is green with a grey background. When he breaks the tile to find the tomb the X has become a faint outline on the floor. (00:27:40 - 00:28:45)
Suggested correction: You still can see one "leg" of the X on the floor, it's only darker than viewed from above because the camera angle and illumination set used.
I think it is meant to be an optical illusion.
The "X" is first shown as a dark green "X" on a beige background. Next, we are shown the same dark green "X" that is barely visible over a green background. I think we are meant to understand that the beige square tiles were lifted away in a cut scene.
I see no reason why they would replace the floor just for the higher shot, it's the same floor throughout the scene. When they enter it's the same floor we see later as they are going into the hole. It's probably not a real marble floor, so they can use a styrofoam or plywood tile that Harrison can lift, one that matches the surrounding tiles. They don't shine as much as the rest of the floor. In the shot up high there is different lighting, so that could explain it. It just appears to be different. Of course, sudden different light can be seen as a revealing mistake.
Suggested correction: Not a mistake, just a different viewing angle.
It's not just the viewing angle. The "X" is gilt-colored and bright but becomes extremely muted, almost a shadow when viewed at the lower angle. Another possible reason for the different appearance is the patterned perimeter. It too appears to be a very different color and muted. It's possible that in post-production, the scene processing done by the cinematography team adjusted the lower angle scene for a change in light levels due to close-ups, the time of day, or some other factor.
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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)
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Plot hole: When Indy is stepping on the letters at the end, trying to spell out God's name, he steps on J, incorrectly, causing it to collapse. When he falls through, however, he grabs onto another letter so as not to fall down. The letters he grabs onto and pulls himself up are an L and a Y, which are not in the word Iehova, so should have collapsed too. (01:46:20)
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Continuity mistake: When Indy is stepping on the letters, he steps on the "O" and knocks the "P" loose. In the next shot, the "P" is intact.
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Visible crew/equipment: When the lead bandit opens the magic cart door to find young Indy running away on the track, a crew member wearing a white shirt can be seen reflected on the door.
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Continuity mistake: When Indy comes face-to-face with Hitler he hands over Henry's diary with its black elastic around the back cover, but in the closeup as the signature is being signed the elastic is around the front cover, then it's around the back cover again when Indy takes the diary back.
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Revealing mistake: When Indy has just climbed the stairs in the library and is leaning over the rail to look at the X below, directly behind him is one of the worst props I've ever seen - a flat fake book case. (00:29:00)
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Continuity mistake: When Indy and Vogel are fighting atop the tank we see the soles of Vogel's boots and there is a red hexagon dot under each heel, but while Indy is hanging off the tank's side mount gun the soles of Vogel's boots are different.
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Other mistake: At one point, Indy rolls off of the side of a tank, and somehow gets a strap around one of the guns pointing out from the sides. The end of the barrel has already been blown open by the rock he shoved in earlier. How did the strap get around that big cluster of metal, and then back again so he could climb back on the tank? (01:34:15 - 01:35:05)
Continuity mistake: When Indy and gang are racing the Nazis to the canyon of the Crescent Moon, Indy looks at the Nazi convoy with binoculars. Close-ups of Indy show the sun is clearly behind him, and the binoculars are in his shadow. But then a glare from the lens alerts the Nazis to their presence. (01:24:25)
Suggested correction: It is not clear that the glare is coming from the binoculars rather than the car behind them (after all, the Nazis target the car with the tank).
They see a glare rather than a car. But anyway, regardless, the problem remains that the sun is behind the car too.
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Continuity mistake: When the tank goes over the cliff, the turret detaches upon impact. But in a subsequent shot, the tank is shown rolling over with the turret still attached. (01:36:35)
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Revealing mistake: When Sallah rides up to the tank to help, just as Indy shouts, "Get Dad!" we can see that Henry is actually lying on protective support, and not lying directly on the moving tank treads. (01:35:40)
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Visible crew/equipment: When Indy tries to reach the Grail his father convinces him to "Let it go" and Henry pulls his son to safety, then in the next wide shot of Indy, Henry, Sallah, and Marcus, we can see a crew member wearing a short sleeved, white shirt and blue pants at the right side of the screen, just before it cuts to Henry's closeup. This crew member can be seen again after the closeup of the knight. (01:57:45)
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Visible crew/equipment: After Young Indy locks the two treasure hunters in the reptile train car he pulls a snake out of his shirt, and just as the third young treasure hunter tackles Indy we can see a crew member's arm and equipment at the left side of the screen, before it cuts to the long shot. (00:07:00)
Audio problem: When Indy confronts the butler at the castle, we hear the butler say, "If you are a Scottish lord then I am Mickey Mouse." But his lips say something else. He actually said "then I am Jesse Owens." This line was changed because they thought that a lot of the audience wouldn't know who Jesse Owens (the black athlete who won at the Berlin Olympics) was. (00:46:00)