Continuity mistake: As Sallah pulls Dr. Jones senior off the tread of the tank, Indy lets go of his whip and turns towards the German colonel and they continue fighting face-to-face. You can see that Indy punches the colonel so that he loses his balance and stumbles backwards. A split second later, the colonel is standing behind Jones with his arm around Indy's neck. They did not have nearly enough time to so radically change positions. (01:32:15)
Other mistake: When Indy and Kazim are fighting on the the boat as it is being chewed up by the giant propeller, the propeller is more than half way out of the water; if the ship was in dry-dock the smaller boats wouldn't be able to go up to the larger ship, on the other hand if it had run aground it would not be running its engines.
Revealing mistake: When Kazim dies, his shirt is all bloody, but there are no holes in it. The bloodstains looks painted on. (01:26:30)
Continuity mistake: The lion approaches Indy from the rear of the circus car and walks towards the center. Next shot, the lion approaches from the back of the car again.
Continuity mistake: The Joneses take off in the Zeppeliner from Berlin. Shortly after takeoff they are over tall snowy mountains but there are no tall snowy mountains that close to Berlin.
Continuity mistake: When Indiana pushes the Nazi colonel out of an open window, if you look carefully, the colonel fell a few metres out of the zeppelin before landing on a pile of suitcases, but when the colonel looks up, the zeppelin is several hundred metres above him, in too short a time to ascend that much.
Continuity mistake: When Indy enters the beach he steps on dry sand and then on the wet part. A frame later, in a wide angle, he is still stepping on dry sand.
Continuity mistake: When Indiana is fighting on top of the tank, he kicks the periscope. It spins counter-clockwise, but the handles inside spin clockwise. (01:32:45)
Continuity mistake: Indy pours water from the grail into his father's mouth. Between two consecutive shots, the position of Indy's hand changes and the direction he pours changes 90 degrees.
Continuity mistake: The shape of the exploded canon barrel on the tank changes a lot in the different shots. Especially between the scene right after the explosion and the scene where Indy is hanging under the barrel.
Continuity mistake: When Indy meets Hitler, there is a shot over Indy's shoulder that shows his lips are slightly parted (relaxed). Cut to Indy's face and he closes his mouth tightly. In the very next shot, with no elapsed time, we see from Indy's shoulder again that Indy's mouth is relaxed as before. (01:10:15)
Deliberate mistake: While Indy is fighting the tank commander, he looks up to see them heading toward a cliff. After the tank goes over, from the time his companions look over the cliff to the time when they walk away and Indy is sitting on the ground, you can see the wind blowing in their faces. But as Indy starts to get up, the wind blows his hat toward him, which it couldn't do as his hat flew off quite a while before, and the winds would have blown it far from the cliff's edge.
Factual error: When Indy and Marcus first arrive in Venice on the boat the dock they alight on to has tactile pavers (for the visually impaired) installed on the platform. These weren't invented until 1965, some 25+ years after the scene takes place.
Suggested correction: One piece of the pier looking different doesn't mean that is meant for blind people, could be just a repair, or decorative, or a coincidence, could be anything.
You can see it is a tactile paver as it has the raised spots.
Continuity mistake: The Zippo lighter on the floor, thrown by Dr. Jones, keeps moving around between shots.

Continuity mistake: In Berlin, Indy grabs his father's book from Elsa's trenchcoat. In the next shot, the Nazi pin on Elsa's tie has disappeared.
Continuity mistake: When Indy shoots the three soldiers on the tank they fall sideways but not off the tank, a few seconds later all three bodies have disappeared.
Other mistake: Indy has been very careful to hide from everyone that Alexandretta is the city where the trail starts, only to give away all the essential info. He did not tell Elsa. Only Marcus (who has the rubbing of the crucial inscription) and Sallah knew. The only way that the Nazis find out is because he tells his dad, in a room which he KNOWS to be wired, i.e. bugged. (The Nazis also find the diary, but we know that the city isn't in there.) Stupidity doesn't quite cover it.
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.
Visible crew/equipment: After Indy's father is cured and the Nazis drop the weapons, he stands up and set lights are reflected on his glasses.
Continuity mistake: When Sallah pulls his rifle and overtakes the Nazis at the Great Seal he is in the forefront with the Nazis in front of him and Indy and his father are behind him. Within the next scene he is now in the background with his rifle resting in his arms.