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 crewmember appears from under the tarp taking hold of the throttle, steering the boat away from the pier. (01:02:40)
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.
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)
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)
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 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)
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.
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)
Continuity mistake: In the temple in the Canyon of the Crescent Moon, Indiana Jones rushes to his father's aid with the Holy Grail filled with water. He is shown pouring every last drop onto his father's wound by turning the Grail upside down, yet another camera angle shows his father clutching the grail to his chest, and there's still a bit of water in it.
Factual error: In this film, and in Raiders of the Lost Ark, the German vehicles are marked with the DAK (Deutsches Afrikakorps) symbol (a swastika over a palm tree.) The DAK and its symbol were not created until 1941 when Hitler tried to rescue his Italian allies in North Africa. (00:56:15)
Factual error: At the end of the classroom scene, Indy gives Marcus the Cross of Coronado. There is a pile of books on Indy's desk that he laters picks up and takes with him. One of the books (with the whitish spine, under the red book) is "Living Egypt", written by Paul Strand and James Aldridge. This book wasn't published until 1969, over 30 years after this movie takes place (1938). (00:15:20)
Continuity mistake: At one point Indiana's father blows the tail of the plane in which they were flying to pieces in an attempt to shoot at the Germans. This is obviously in a close-up studio shot. The next moment the plane is shown flying in the distance with its tail still intact. (01:17:45)
Revealing mistake: During the "What happens at 11 o'clock?" joke, Jr. counts down ("12, 11, 10!" or something like that) with his hand, matte lines surround his hand to the point where it almost disappears.
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.
Continuity mistake: While Indy and Marcus are in the apartment in Venice, after Marcus says, "He knew everything, except the name of the city", he slams the diary shut. Then it's shown open, and he closes it a second time. When Indy takes it, it's once again open. (00:42:15)
Other mistake: When they go under the coffin you see the flame of the torch in the background. How could the flame go under the petroleum with them without either setting light to the petroleum or being extinguished itself? (00:35:55)
Factual error: Jehovah is spelled Iehovah in Latin. However medieval languages (Latin or vernacular) had neither official rules nor the letter 'J'. It was not until the sixteenth century that the French humanist Pierre de la Ramée proposed to use the 'J' as a separate letter distinct from the 'I'. The 'J' could therefore not have been a trap at the time the test was constructed. (01:46:20)
Suggested correction: The concept of what looks like a "J" was sometimes used when writing numbers in Roman numerals, originating sometime in the Middle Ages. While "J" did not exist as a letter at that time, what visually looked like a "J" did exist at the time the Trial was made. The purpose is to weed out those who can't spell the word of God...there's nothing implying that whoever created it meant for all the tiles to be understood as just letters.
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.
Factual error: When Indy and his father are being chased by the German soldiers on motorbikes they can be easily identified as modern lightweight Japanese trailbikes (Honda XL500S from the '70s/'80s), not the big heavy bikes the German army used. Furthermore, the sidecar used by Indy, instead of being a BMW supplied to the Wermacht, is a Soviet copy (Dniper or Ural) produced in the 1980s, disguised to look like a BMW. (01:03:35)
