The songs that Val Kilmer sings in the film are actually performed by him and were featured on the film's soundtrack released in 1984 under Kilmer's character's name Nick Rivers.
The waiter who asks if Nick is ready to order, ostensibly in German, is actually using the Yiddish curse "Go take a shit in the ocean."
The prosthetic that Peter Cushing wears in the bookshop scene involved the making of a cast of his face to fit it to. This same cast was used 30 years later by ILM to build a CGI model of his face for Rogue One: A Star Wars Story (2016).
When Nick Rivers is in jail, a head shot of Cher can be seen on the wall of the cell. Val Kilmer was dating Cher at the time.
When Nick is leaving the East German checkpoint by train, a voice can be heard over the PA saying, "Der Zug der jetzt auf Gleis 3 steht... hat uns alle überrascht!", which translates to "The train that is now on platform 3... surprised us all!"
Jim Abrahams, David Zucker, Jerry Zucker: as the German soldiers who show up at the door of the prop room to arrest Nick.