Richard hits Tommy in the face with a 2x4 when they get into a fight. This board was nowhere in sight in the long shot of them getting out of the car, and not enough time passes between shots for Richard to have run off, found it, and returned.
Near the beginning, when Tommy's dad shows Tommy his office, the door reads "Thomas R. Callahan III." At the end of the movie, when Tommy is being introduced as the new President, he is introduced as "Tom Callahan Jr."
When Tommy bends the car door back into position at the gas station, it is visibly dented and crumpled along the edge. Immediately after, Richard goes to open the door, it falls to the ground, and is in pristine condition.
Richard hits Tommy Boy across the side of his face with a board. It is revealed, during the lunch scene at The Cluck Bucket, that this left a huge bruise on Tommy Boy's face. Yet just a short while later, when Tommy makes his first successful sale to Ted, there is not a trace of any bruise on his face (nor is any sign of the bruise seen at any time later in the movie).
While in Zalinsky's Board Room with the TV crew, Tommy gets Zalinsky to give his "I make car parts" line. As Zalinsky starts, he opens his shirt and undoes his tie, next shot his tie and shirt button are closed, then the next shot they are open again.
During the last scene on Tommy's boat, there is no rope connected to the pulley wheel at the end of the boom of his sailboat. In the next shot when he sails away, there is a rope attached to it (this is necessary to be able to sail).
In the scene where Tommy wrecks and lights the toy car model on fire, the front axle and wheels are missing from the table in some shots, and the blue car is set on the table in different spots in some shots as well.
At the end, when Tommy is sitting in the boat, we can see as the camera moves that the boat is only in about two feet of water. In the next scene, the boat appears to be in the middle of the lake, and in the next scene he is close to shore again.
In the scene where Tommy and Richard stay at the Deer Glen Hotel, Richard makes a comment to Tommy about the moths inside of the hotel room. If you look closely you can see a member of the crew's hand throwing moths in the air at the base of the lamp behind the end table.
At the beginning of the movie, Tommy rushes to take his American History Exam. The first question is "James Madison, Thomas Jefferson and _____ Hancock were all Framers of the Constitution." (Tommy writes Herbie). However, a University History Professor would know that while Madison was a Framer, Hancock and Jefferson were not. The latter two were not at the Constitutional Convention.
At the end of the movie when when Calhoun starts the motor and the cooling fan starts it pushes the toupee of Richard back, however cooling fans of a motor do not push air, it pulls air through the radiator to cool the water so the toupee would not get pushed back.
Early in the movie Richard claims to have dropped a 440 6-Pack Magnum engine into his Plymouth GTX. In all later scenes with the hood up or missing, it shows that the car is actually equipped with a standard 440 4 Barrel, as indicated by the "440 Magnum Four Barrel" Air Cleaner. The Magnum and the 6-Pack are two different packages for the 440, impossible to confuse with one another. 440 4 Barrel engines are equipped with a single four barrel carburetor and round air cleaner, where 6-Pack engines are equipped with three two barrel carburetors and a massive oblong air cleaner that covers the entire length of the intake.
At the beginning of the airplane bathroom scene, Richard's double can be clearly seen entering the bathroom - the double's hair is much more gray than Richard's. Then, visible through the crack in the door, the double can be seen to go left as Richard comes back of the bathroom.