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)
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: 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)
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)
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.
Continuity mistake: Inside the zeppelin when Indy and his father are talking at the table, notice the white handkerchief that moves all around the table. (01:13:50)
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)
Continuity mistake: In the speedboat scene where Dr Schneider rescues Indy and Kazim from the boat that is chopped up by the propeller of a ship, she driving the speedboat with the steering wheel on the right-hand side. Seconds later as she drives the speedboat away, the steering wheel is on the left-hand side, obviously a flipped shot. When the speedboat slows down to allow Kazim to alight at the jetty, Indy is at the wheel which is now back on the right-hand side. (00:39:55)
Continuity mistake: In the beginning, when Indy is fighting aboard the Coronado ship, the very first time we see Indy in this scene (before we see him get punched), he is bleeding from the right side of his mouth. Throughout the remainder of the scene, he is bleeding from the left side of his mouth. (00:11:50)
Continuity mistake: When Sean Connery drops his lighter while trying to burn the ropes, the lid is flipped to the right. When he blows on it the lid then changes position slightly relative to the pattern on the rug, then when the rug catches fire the lid's suddenly flipped completely to the left. (00:56:40)
Factual error: As Indiana Jones opens the door to Dr. Jones' house, note the light switches just inside the front door. They are the kind developed in the late 1970s and still in use today, not consistent with the film's setting of 1938. (00:22:30)
Continuity mistake: When Indy takes the mail out of his pocket in the beginning of the film, he keeps the "Grail Diary" in his hand, but puts the letters down twice. Once before he opens the package and once after with the wrapper. (00:23:00)
Continuity mistake: As Indy swings through the window to rescue his father, he gets hit on the head by a vase. His hat is all crumpled and tipped quite far back, then in the very next shot his hat is neat and level.
Continuity mistake: When Indy and the protector of the Grail are in the boat about to be destroyed by the rotors, there are panels on the top of the boat that pop up and are destroyed when the boat is moving backwards into the rotors. In the next shot, they are intact, then in the next shot, they are destroyed again. (00:39:20)
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)
Revealing mistake: When Indy is clinging to the side of the tank turret in the desert, to the right of the tank are tire tracks from the vehicle shooting the scene. They appear to just start at a set point. There was likely no way to hide this.
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.