Continuity mistake: When Indiana is showing Marcus the Grail Diary in his fathers home, he closes the book 4 times in a row without opening it - first 3 times inside the room, and then once on exit.
Other mistake: When Indiana Jones first enters the castle and speaks with the Nazi officer he is mouthing the Nazi's dialogue as the Nazi is speaking. (00:45:55)
Continuity mistake: When Indy and Elsa escape underwater, and surface from the tunnel, Indy's suit should be drenched, but it only has water stains. Then during the speedboat chase, their clothes are completely dry.
Audio problem: In the boat fight scene, Kazim says something like "If you don't let me go, we both die" and Indy shouts "Then we'll die." But if you look at his lips, he's saying something else.
Continuity mistake: When Indy and his father are being pursued by German soldiers on bikes, right after they break through the barrier, the barrier guard comes out of his post and waves the pursuing German soldiers to slow down in the background. In the next shot he's no longer there, and the barrier's quite far away, then in the following shot he's back, and Indy's not so far from the barrier again. (01:04:35)
Continuity mistake: Indy falls off the side of the tank and gets his shoulder strap caught on the gun - this should not be possible because his shoulder strap is underneath his jacket, as can be seen when he gets back up onto the tank.
Continuity mistake: In the catacombs underneath the library, when he find the "X" on the wall, he is shown ramming the wall twice to break through. The first time he rams it, however, the wall looks as if he had already had a few goes already. (00:31:55)
Continuity mistake: At the very end there is a large near circular blood stain on the left sleeve of Sean Connery's shirt. When they ride off, the stain has disappeared completely.
Continuity mistake: When the military caravan with Brody is driving through the canyon towards the Joneses and Sallah, the sun is behind Brody (discernable by the shade). But when Indy is looking down on them - from the front - the sun is also behind him.
Continuity mistake: When Indy takes four horses before battling with the tank, three are black and one is bay (brown with black legs, mane and tail). Sallah is riding a chestnut. When they ride to the Grail Temple, there are two black horses and two chestnuts. Where did the second chestnut come from? (01:28:00 - 01:39:25)
Audio problem: When Indiana is fighting one of the bad guys on top of the rhino's car on the train, the rhino sticks his horn straight between Indy's legs. Indy says, "Holy smokes" but if you look, he actually says something else. (00:07:35)
Factual error: When Indiana's father uses the plane's gun, he shoots out the tail. This was not actually possible, as the planes had a safety mechanism to stop people doing this. In fact we can see that the Lewis gun he is using is mounted on a Scarff ring, which prevented the trigger of the gun being activated when the barrel was pointed at the aircraft's tail surfaces. (01:17:40)
Continuity mistake: The sky colour/condition changes between the shots of Indy and his father in line to get on board the Zeppelin and the scene just before. In one, it is cloudy, and in the other it is a clear blue.
Continuity mistake: In the temple in the Canyon of the Crescent Moon: After Harrison Ford cures Sean Connery by pouring water from the grail over him, Sean stands up with the grail in his hand. In the next shot, however, Alison Doody picks the grail up off the floor. (01:55:50)
Revealing mistake: The biplane sequence is obviously filmed in a studio close up. Despite the fact that they are miles high and swooping around at high speeds, neither Indy's hat, nor his Father's, blow off or even flap in the wind. (01:16:55)
Revealing mistake: When Indy is being chased on the motorboard the guy behind tries to shoot him with a machine gun. If you watch the machine gun carefully it you will see it starts to disintegrate in the guy's hands and the barrel drops off. (00:38:45)
Revealing mistake: When fire sweeps through the catacombs under the church, you can see that the "rats" are completely fake. At the far left of the screen (widescreen) you can see several fake rats just hanging on the wall and not moving at all. In the "special features" section on the DVD, Steven Spielberg admitted that the rats were fake - he couldn't bring himself to roast live ones so he could not make this scene appear realistic.
Factual error: There are no catacombs in Venice! There are also no natural petroleum sources - but that seems secondary to the fact that the entire city is just below sea level - there are no sewers, no subways - nothing underground.
Factual error: In various scenes, Dr. Elsa Schneider (Alison Doody) is wearing the M43 (Mountain Cap). That cap was introduced in 1943.
Answer: Henry says, in response to Indy asking if he remembered the details of the trials: "I wrote them down in my diary so that I wouldn't have to remember." So, obviously he did NOT know them by heart. Also, as the other answer says, they didn't want the diary to either be in the Nazis' possession or be burned.