After Feynman visits the Marshall factory, he records notes on an electronic typewriter. One note reads (electronically) "Cost versus reliability" which we then see printed as "Cost VS reliability". The typewriter would not abbreviate "versus", but would print verbatim off the screen.
(at around 42 mins) Typed notes show "bizarre", while typewriter / word processor display shows misspelled "bizzare".
When Rogers gives Feynman a "ride" to Alabama, throughout the flight, you can see Dick's mouthpiece move several positions, especially from the different camera angles.
When Feynman visits the Shuttle factory, the flight deck interior is clearly that of a Boeing 747-200. You can see the flight engineer's panel and the four throttles on the centre console, also the windscreen layout gives it away.
Each orbiter shuttle was built differently. In the beginning stock launch footage used to represent the telecast for the STS-51L launch (the loss of Challenger) particularly at the moment of main engine start is clearly that of a different vehicle. The tail does not match that of OV-099 Challenger, but rather one of the newer vehicles that used fewer black silica tiles (most likely OV-104 Atlantis).
When Feynemen is visiting the orbiter during his inspection, it is shown with the NASA "Meatball" logo, which was not used on any orbiter until 1998.
Emergency Exit signs at Marshall Flight Center in Alabama are green. Green was not adopted as the international color for exit signage until the turn of the 21st century. In 1986, exit signs in the United States were almost exclusively red while green exit signs were exclusive to the United Kingdom.
When Feynman vomits in the men's room, he flushes the toilet on a modern automatic-flush toilet. These toilets did not exist in 1986.