Continuity mistake: Indy empties the fridge hurriedly and throws away all of the trays, then pulls away the last drawer below. A shot later from a different angle, he is throwing away the penultimate tray and pulling out the drawer again.
Sacha
Continuity mistake: When Indy exits the fridge after the blast, he leans on both of his arms and faces a gopher. From a different angle, he is leaning on his left arm only.
Factual error: In the church, you can see modern brass panic bars on the doors.
Factual error: During the motorcycle chase seen (which was filmed in New Haven, CT) one can see modern yellow traffic lights covered in burlap in a few of the scenes.
Visible crew/equipment: As they are running away from the ants, you can see a rig of stage lights under the jeep Col. Spalko's crew was driving.
Continuity mistake: After Mutt's bike knocks down the "Better red than dead" banner it falls sideways; yet when the chasing car arrives, the banner is magically up again, ready to fall horizontally on the car's windshield.
Revealing mistake: During the campus chase scene, Mutt skids his bike and turns left to drive under a stone arch. Multiple skid marks are seen on the pavement from previous shots.
Continuity mistake: The military car racing against the young kids has its right windshield covered in dust, except for the wiper's track. A frame later the glass is clean.
Continuity mistake: Before Indy finds the crystal skull, the mummy's head has a piece of cloth on its forehead. In a wide shot immediately afterwards, this cloth disappears.
Continuity mistake: During the ants scene, Indy is about to grab a stick to the left of his hat. In the close-up the stick and the hat have swapped sides.
Continuity mistake: At the diner, Mutt places the photo on his hat, yet it swaps from standing out to tucked inside, between shots.
Continuity mistake: When Mutt first shows up, he is riding his bike between two train wagons, or between a train and a railing next to the station. This changes back and forth with no logic at all.