When Quaid shows the new guy where to safely hold parts on the robot they are assembling, in the next shot the new guy doesn't seem to take any notice and puts his hand in the unsafe position again and continues working.
If The Fall is free-falling and assuming no resistance, the occupants would be weightless for the entire trip. However people were shown in normal gravity except for the midpoint flip. At 1-G the trip would take 42 minutes (a lot more than the 17 minutes as suggested in the movie), therefore the capsule has to be accelerating/decelerating at higher rates.
For The Fall to travel the diameter of the Earth in 17 minutes, it would have to maintain an average speed of over 45000 kph (28000 mph). At this speed, it would be impossible for anyone to survive standing on the outside, because of both the airflow and the high temperatures resulting from friction.
To travel 8,000 miles in 17 minutes the Fall has to have an average speed of 28,235 mph. At those speeds the ship would have to be traveling in a vacuum or the Fall would burn up with the friction of the air.
The music credits mention only Beethoven's Piano Sonata No. 17, but the first part Quaid played on the grand piano is actually from the first movement of Beethoven's Piano Sonata No. 14 "Moonlight" C# minor op. 27/2 - only the second piece is from the opening of the third movement of Ludwig van Beethoven's Piano Sonata No. 17 "Tempest" in D minor, Op. 31, No. 2.
When Carl leaves the rebel base in the wastelands he is wearing a respirator, but he enters the "helicoper" without passing through an air lock. Thus he would be exposed to what ever the respirator was meant to protect him against.
The Arabic characters written on the top left side of the passports make no sense. They are single random letters, whereas in Arabic, words are mostly written attached together. The same goes for the Persian language.
As Hauser plays the excerpt of Ludwig van Beethoven on the piano in his apartment to unlock The Key, there are piano keys which either remain depressed after his fingers leave the key or, more likely, press themselves (like a pianola) revealing that Colin Farrell is not actually playing the excerpt for real.
The goof items below may give away important plot points.
In "The Fall" when Quaid flies in with a helicopter, he enters a room where he sets the explosives. Right before he enters that room, you can see from the window behind him that "The Fall" is moving. However, the release of it is shown after Quaid enters the room.
When Matthias is shot, you can clearly see black cloth on the robot's neck and shoulder areas where mechanical pipes and rods are supposed to be. They are visible in wide shots, however the black clothing can be seen on close-ups on the robots standing next to Melina.