Continuity mistake: After Indy and Elsa exit the catacombs through a sewer and start to run away, a woman in pink turns her head to watch them leave and then turns around to see Kazim's men run after the couple. A frame later, from a new angle, the woman's head is turned 45° facing downwards.
Continuity mistake: Sallah rescues Dr. Jones from the tank and stops by the cliff. On the next wide angle he's nowhere around.
Continuity mistake: Indy stops his sidecar at a crossroad, meters ahead of a house. When the angle changes he is now several meters behind.
Continuity mistake: During the crashes by the border control the road keeps swapping from sunny to shadowy back and forth in less than 3 seconds.
Continuity mistake: When Marcus watches the carbon copy of the shield, his ice-bag is on the table, but when the angle changes he has it next to his head and seconds later he puts it on the table. Also, the ashtray moves backwards between shots.
Continuity mistake: On the tank, Indy is punched and the gun falls inside. His hat also falls off, but a shot later it's back on, perfectly tucked in.
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.
Continuity mistake: After Dr. Jones fires at the jeep two soldiers fly towards the tank, but when the angle changes they're falling away from the tank.
Continuity mistake: Inside the temple, when Indy is at the cliff, in the close-ups the rocks next to him are round shaped, but in the wide angle they are straight and stripped.
Continuity mistake: By the cliff, Dr. Jones hugs Indy, whose jacket is down at waist level. When Dr. Jones leaves, Indy collapses on his knees and now the jacket is perfectly placed over his shoulders.
Continuity mistake: By the border, when the German soldier's been knocked off by Indy and his bike rides alone, he suddenly disappears from the side of the road. (01:02:10)
Continuity mistake: When Indy leaves the steering wheel to stop the baddie on his motorboat, he passes by a metal turret twice in less than half a second.
Continuity mistake: During the fight on the tank, Indy punches a German who falls on the wheel and whose hat falls near the turret. When the angle changes the hat is in front of him, but when he falls off the tank the hat is gone.
Continuity mistake: The plane explodes at the exit of the tunnel. A frame later the exit is clean and the small remains are scattered around.
Continuity mistake: When Indy and Elsa encounter the rats, hundreds of them block the way, making Indy lift up his legs. However, a frame later, when Elsa starts to walk, those rats have disappeared.
Continuity mistake: After escaping from the turning blades, Indy's hat swaps from clean to filled with spiderwebs. Though there are 4 seconds in between, he didn't move from the previous spot, nor were there any webs around.
Continuity mistake: While in the train station in Iskenderun, Marcus passes by the same spot and the same people twice (watch the man with the hen on the left).
Continuity mistake: In the catacombs, Indy finds the tomb and blows dust away from the shield, cleaning the sides of the engraved cross, but leaving the rest very dusty. A frame later, the whole cross is spotless.
Continuity mistake: While the propeller chops the boat, Indy grabs Kazim and lifts him up saying, "This is your last chance." But from the opposite angle Kazim is still seated.
Continuity mistake: When young Indy discovers the thieves in the caves, the pot on the right swaps from laying horizontal on the ground to slanted and back to horizontal, depending on the angle.
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.