Ability over Appearance: Only a couple of the actors who played the German terrorists were actually German and only a couple more could speak broken German. The actors were cast for their menacing appearances rather than their nationality. Nine of the twelve were over six feet tall.
Beam Me Up, Scotty!: Hans' quote on Alexander the Great. It's actually "Alexander of Macedon, who, upon hearing that there were other worlds, wept that he had not yet conquered one", though this would fit in with Hans' posturing and Delusions of Grandeur.
Completely Different Title: The original title, an English idiom, is hard to translate correctly, as it would sound like "It is hard to kill him" or "He dies slowly". That's why we have:
Czech Republic and Slovakia: "Lethal Trap" (which may often confuse people as Lethal Weapon, as the Czech title of said film was translated literally)
France and Italy: "Crystal Trap"
Hungary: "Give your life expensive". The title of the sequel is "Your life is more expensive", and the third part is "The life is always expensive".
Norway: "Aksjon Skyskraper"
Poland: "The Glass Trap" (which sounds and fits very well in the language, but does not make sense for sequels).
Portugal: "Assalto ao Arranha-Céus" ("Skyscraper Heist").
Russia: "Hard Nut" (to crack).
Spain: "Crystal Jungle" (same problem as in Poland).
The Finnish version is an interesting case: the original title was "Vain Kuolleen Ruumiini Yli" ("Over My Dead Body"). While the translated title is fairly close to the original, nowadays the original English title is used to refer to the movie (or, alternatively, the series as a whole).
Defictionalization: You can buy gray sweatshirts that say "NOW I HAVE A MACHINE GUN. HO HO HO."
Divorced Installment: The film was was based on a book, Nothing Lasts Forever. This book was a sequel to The Detective, which had its own popular film adaptation in 1968, starring Frank Sinatra. When Sinatra declined to be in a film sequel, it was quickly retooled as a stand-alone work. Rumors that it was briefly intended as a sequel to Commando was debunked by co-writer Steven de Souza, though Arnold Schwarzenegger was also considered for the lead.
Enforced Method Acting: Alan Rickman was told that he was going to be let go on a count of three. They dropped him on "two," and the look of panic on his face is definitely not acted; one is not surprised to learn that he was extremely angry after that shoot was over.
In an episode of Stargate SG-1 entitled "Bad Guys", where the team find themselves mistaken for the villains of a "Die Hard" on an X scenario, the leader of the team refers to the bumbling security guard trying to catch them as "a bit of a John McClane". When one of the team members (Earthling, born and bred) doesn't get it, the resident alien of the group namedrops the film.
Running the Asylum: Arguably one of the biggest fans of the Die Hard franchise is... Bruce Willis. He apparently feels very strongly about not insulting the viewers' intelligence, and while his self-appointed "gatekeeper of the franchise" role makes directors cry, among other things it resulted in the villain of the fourth film actually getting a fully fleshed-out backstory (rather than what Willis described as a scene of "MySpace and cheerleader porn jokes").
In a way, the first movie also counts: the book that it's based on was a sequel to The Detective, which was made into a film starring Frank Sinatra in 1968. Die Hard didn't come out for another two decades.
Technology Marches On: John McClane's inability to contact the outside causes him some problems initially, as he's forced to use a captured radio to try calling the police. If he had had a mobile phone, the movie would have gone much differently. Humorously, Argyle spends most of the film luxuriating in the fact that he can call his friends on his limo's car phone.
The line "Alas, your Mr Takagi did not see it that way, so he won't be joining us for the rest of his life." was an ad lib; Alan Rickman also improvised the bit of helping himself to the party buffet while saying this. Much of the scene where Hans tries to use a fake American accent to pass himself off as a hostage to McClane was apparently ad-libbed after the producers discovered that Rickman could do a good American impersonation.
The shot in the elevator shaft where McClane drops from the vent he was aiming for, but manages to grab the next one down was an accident on the part of the stuntman that made the final cut anyway.
Al Leong improvised the bit where he takes a candy bar from the concession stand before fighting the SWAT team, as he felt the scene could use some comic relief.
It is often said that Bruce Willis' lines during the scene when he pulls the glass out of his feet were ad-libbed. Indeed, it is said that upon learning this, Terry Gilliam cast Willis as the lead in 12 Monkeys. However when comparing the original script, it appears that Willis only veered very slightly from the original written dialog.
Willis has claimed that the iconic "Yippee ki yay, motherfucker" was ad-libbed, playing off Hans Gruber's cowboy taunt. Willis did not expect that line to make the cut.
Ellis' line "Hans... Bubby!" was ad-libbed. Alan Rickman's quizzical reaction was genuine.
The references to VCRs, and the fact that John McClane seems really uncomfortable using the computer monitor at the front desk of Nakatomi Plaza.
Neither John nor anyone else in the building can call out after the bad guys cut the building's phone lines, because no one (other than Argyle) has a cell phone.
When Sgt. Powell responds to dispatch about John's call from the Nakatomi Plaza, you can see a gas station price sign in the background: Unleaded 77 cents, Regular 74 cents. Even the existence of Regular leaded gasoline is itself an example after the banning of leaded automotive gasoline in the 1990s.
During the beginning of the film, John having his service pistol with him on his flight to Los Angeles. Unthinkable in the modern day, but perfectly legal until 1994.
This is the big one: Nothing Lasts Forever, the novel in which this franchise started from, was a sequel to the novel The Detective, which was adapted to film starring Frank Sinatra. Sinatra was 73 at the time, but a clause in the first movie's contract meant he had a right to reprise his role as Joe Leland in the sequel before it could be offered to anyone else. In other words, Frank Sinatra technically was and would have been the original John McClane, which actually would have fit pretty well in the adaptation, as the source novel's hero was an old, retired cop who found his inner badass during a terrorist siege.
Arnold Schwarzenegger was the production team's choice for John McClane, and following Sinatra's above-mentioned rejection, the script was written to accommodate him. Once Arnie too turned the role down and Bruce Willis was cast, the movie had to be considerably altered to fit the latter's acting style.
Alan Rickman nearly passed up the role of Hans Gruber, which ended up being his first film role. He had only arrived in Hollywood two days earlier and was appalled by the idea of his first role being the villain in an action film.
The original script called for terrorists to hijack the building, and for a super-hero cop to stop them. McTiernan modified the script to change the bad guys into robbers pretending to be terrorists so that the audience could enjoy their intention of grabbing a load of money. He felt having terrorists as the villains would make the movie less enjoyable and give it a political angle, which he wanted to avoid. McTiernan also changed the hero, John McClane, into an everyday, flawed man that rises to the occasion in dire circumstances. He felt the audience would identify more with him than with a "super-cop".