When Harry takes the pictures of Elliot and Helen from inside the vent, he is at a distant angle. The pictures he shows to Elliot, later, were taken from a closer point of view and from the side.
When Harry Crumb is undercover as the Vice President of Quality Control at the spa, he smells the mud and gets some on his nose. A few seconds later, the mud is gone and his nose is clean.
While following and watching Eliot toss the briefcase in the dumpster, the license plate on Vince's car changes styles between close and wide shots.
When Harry first eyes the paper shredder in his new office, the collection bag is empty. After feeding a small card into the machine and getting his necktie caught in it, the bag is full of paper scraps.
As the plane is taxiing from the gate, the pilot announces that they will be flying at 32,000 feet. In 1989, that altitude was unusable. Above 29,000 feet, altitudes were assigned in 2,000 feet increments. After 29,000, you went to 31,000, then 33,000 and so on.
While speaking with Mrs. MacIntyre, oranges fall from the tree in Harry's office. When the first one falls, it would appear that a crew member's hand briefly appears, pushing the orange off the table, but watching the scene in slow motion/frame advance reveals that the "hand" is actually the leaves of the orange.
When Elliot is tied up in the hanging chair and Harry hits him with the door, Eliot swings around and his feet hit one of the equipment racks, causing the rack to wobble as if it were just a hollow shell.
It appears that Jennifer's kidnapping scene was a second take. As the bowl of mud is knocked over, quite a bit of mud was already on the shoe and none was spilling from the bowl while on the way down.
When Harry drives off in the green car from the rental agency in Los Angeles, it clearly has no windshield wipers.
As Harry falls out of the AC vent and onto the apartment manager, knocking him out, the side of the apartment manager's face is different, revealing that a stunt double was used.
During that car chase when Harry gets his car hooked onto the car in front watch the scene closely because it's clearly a stunt driver doing the driving and Nikki is no longer in the car with him, if you go slow frame by frame it shows this.
The ransom money is paid in $1,000 bills, which went out of circulation in 1969. While $1,000 bills were still available in 1989, they were very rare and cost more than $1,000. No one would've paid a $10 million ransom in $1,000 bills at that time.
When Nikki calls out to her father, to let him know that Harry arrived at their house, her mouth is saying something other than "Hey, Dad!".
Just after Harry Crumb falls out of the air conditioning vent, his camera also falls out. Hands are visible in the metal vent, dropping the camera.
During the car chase, just after Harry's car breaks free, camera mounting equipment is visible on the driver's side of Vince and Helen's car.
When Detective Casey begins following the ransom money, camera and crew are reflected in the vehicle windows as he drives away.
When Vince and Helen drive toward the dumpster where Eliot parked, part of the camera crew is reflected in the rear-view mirror, just before Vince's face comes into view.
Vince would never have been able to get a handgun through the metal detector at the airport, let alone the fact that when he confronts Eliot and Helen as they embrace and holds Eliot at gunpoint, there is not a single person around to see him pull a gun. Airports are never that deserted.