In the Mayan temple, Indy comments "I have a bad feeling about this." This line was a running gag in George Lucas' Star Wars saga, and was used by Harrison Ford at least once.
When in the tent and forced to face the Crystal Skull, Indy dismisses the skulls as being from "Saucer Men From Mars". Indiana Jones and the Saucermen from Mars was actually an early title for the film.
Among the artifacts that fall out of the crates Spalko and gang push aside in the warehouse scene is Moses's staff. Property master Douglas Harlocker modelled it after the staff wielded by Charlton Heston as Moses in Ten Commandments. Incidentally, the Indiana Jones videogame, 'Indiana Jones and the Staff of Kings', is about Indy's quest for Moses's staff.
Before production started and while the movie was still being planned, Harrison Ford was shown the original Indiana Jones costume from the first movie, and he has said he was pleasantly surprised that he could still fit into it perfectly despite being much older, so all doubts he might have had about "fitting into the role" went out the window.
The scene where Indy escapes a nuclear blast by hiding in a lead-lined refrigerator spawned the phrase "nuking the fridge", which is analogous to jumping the shark, an idiom to describe when a TV series has passed its creative peak and its overall output has begun to decline.
The student in the Marshall College library, that Mutt and Indy almost run over at the end of the chase, is Chester "Chet" Hanks, son of Tom Hanks.