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)
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.
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)
Audio problem: When Vogel is holding Elsa hostage in the castle, he points his gun at Elsa's neck and says "Enough, she dies." If you look at Indy's shadow on the wall, you can see Indy is mouthing the word "Wait." In the next shot, he is saying the word again, this time with audio intact.
Audio problem: Just after young Indy escapes the bandits on the train, he finds the magic cart and says "Magic" to himself, but a close look will show his mouth doesn't move with the line.
Audio problem: When the temple floor cracks, Indy holds Elsa from falling into the void and she gives a very loud cry, but her mouth is closed.
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.