While at dinner for the first evening with Ben and Eve, Chauncey's wine glass fills and empties within seconds.
When Chance is bumped into by Eve's car, his left leg is injured. However, it's his right leg that is examined and treated by the doctor at the Rand mansion.
When Eve enters Chance's room as he is eating breakfast in bed, the napkin beneath his chin changes position back and forth between shots.
Chance tries to use the bed's adjustment device to change the TV channel. In doing so, the pillows fall forward onto the bed. The next time he passes the bed, the pillows are back in place, upright.
When Eve and Chance first enter the Rand estate by limousine, the car turns left inside the gate, but when they arrive at the front door, they are approaching from the right.
When Chance is watching himself on the large screen in the store window display, he uses his home remote control to try and change the channels on that set, but it controls another TV instead. However, the remote is of the earlier "ultrasonic" technology, and these sound waves do not pass through glass at the required strength needed to work. Those born before this movie came out may recall that jiggling keys would make a remote-controlled TV unwittingly change channels or turn off and on. At about this same time, the now common "infrared" RCs were coming out, albeit at outrageous prices. These signals can, in effect, pass through glass, despite some losses due to reflection. For the scene to be technically correct, an IR remote should have been used, which would not make those "ringing bell" metallic sounds.
After the President leaves the Rand house, the table clock in the hall that Chance is watching indicates 10:30, but it chimes twelve times, even though he's been told several times that they are very accurate. [Note: the statements on accuracy refer to the keeping of time; an error in the number of chimes does not necessarily affect the time keeping. However, it is logical that taking pride in the clock would include ensuring that it functioned properly in all aspects.]
When the President is shown aboard Air Force One asking aides about "Chancey Gardner", immediately prior is shown a twin engine Boeing 737 taking off. A 737 has never been outfitted as a presidential plane. It would have been more accurate for a Boeing 707 to be taking off.
The Boeing 737 used as Air Force One is an American Airlines commercial jet.
When Chance touches the "dead" old man's forehead, the old man's chest rises and falls as he breathes.
Once Chance's character became nationally famous as "Chauncy Gardiner," the mass media would certainly have dug into his past to determine when and where he was born, where he grew up and went to school, who his parents and friends were, etc. The media would have found no history of "Chauncy Gardiner" and Chance's restricted and isolated upbringing, simpleminded personality and lack of education would have been revealed almost immediately.