Beam Me Up, Scotty!: As the first phase of the MCU culminated with The Avengers, many fans remember back to Nick Fury in The Stinger wanting to talk about "the Avengers Initiative", however the actual line is "I'm here to talk to you about the Avenger Initiative." (No "s".)
Jon Favreau cast Robert Downey Jr. after seeing Kiss Kiss Bang Bang. He also believed that Downey's past made him an appropriate choice for the part and that he could not only make Stark a "likable asshole," but also portray an authentic emotional journey, once he had won over the audience.
Favreau cast Terrence Howard as Rhodey because he believed that he could play War Machine in a sequel. Obviously, that ended up not being the case.
Jeff Bridges grew a goatee and shaved his head, which he said was something he'd always wanted to do.
Peter Billingsley has a cameo as the lead Sector 16 technician (a.k.a. the guy Stane chides for not being able to reverse-engineer the Arc Reactor). He had his hair shaved to appear as male pattern baldness.
Averted with Robert Downey Jr.. He's in shape, but he's not the absolute beefcake that pretty much every other MCU lead would be in later movies.
Harpo Does Something Funny: Apparently a good deal of the dialog was like this; not just Tony Stark's. It's reported that most of the script was a brief summary of what the actors needed to say, and from there they were allowed to pretty much improvise the finer details, which is why the dialogue feels a lot more naturalistic. This is how everyone discovered Robert Downey Jr. is a witty bastard, hence Iron Man's new characterization as a Deadpan Snarker. Jeff Bridges described the experience as a if he were working on a $200 million college film.
The Other Darrin: Famously, Terrence Howard plays Rhodey in this film, to be replaced by Don Cheadle for the rest of the franchise. Less famously, Gerard Sanders portrays Howard Stark in a photo montage early in the film, to be replaced by John Slattery in the later films (except Captain America) and Dominic Cooper in the TV shows (and Captain America).
The film had a lengthy production cycle that dates back to the early 1990's. Jeff Vintar and Stan Lee pitched a story for Fox that would have had M.O.D.O.K. as the main villain, and Quentin Tarantino was approached as the director. Due to having too much on their plate with X-Men and the studio's other Marvel movies (as well as Tom Rotham's rumored dislike of superheroes), Fox ended up selling the rights to New Line.
Another draft had Crimson Dynamo as the villain, aspects of the character were incorporated into Whiplash in Iron Man 2 by virtue of making him a Russian named Vanko, like the original Dynamo.
According to Jeff Bridges, Stane was originally supposed to survive the final battle against Tony, with the heroes opening up the destroyed Iron Monger suit to find that there was no corpse inside. Presumably this would have poised him to return for future movies.
There was originally a detailed explanation of the suit's HUD, but test audiences turned out to quickly be able to make sense of it, and were actually more confused by the explanation, so it was all taken out.
Samuel L. Jackson recorded an alternate version of the now-famous post-credit stinger, which featured Nick Fury referencing "radioactive bug bites and assorted mutants" during his speech to Tony. It's unknown if this was meant to imply that the then-recent Marvel films from other studios like Sam Raimi's Spider-Man trilogy or Fox's X-Men movies existed in the same continuity as Iron Man, but the lines were presumably omitted due to legal reasons (though ironically Spider-Man entered the films directly later on, and the Disney/Fox deal opens the door for the X-Men to appear). This alternate scene sat on the cutting room floor for over a decade until it was released as an extra in a limited edition box set of the entire Infinity Saga.
Even the ending of the film was an ad lib! Originally the MCU would have proceeded like the comics, with "Iron Man" being passed off as Stark's bodyguard, but Robert Downey Jr. decided at the last minute to ad lib Stark throwing away the script and revealing the truth. Kevin Feige took a chance on making such a dramatic change to the canon, which encouraged their later experimentation and the creation of a more original franchise.
The Mark II's icing problem was originally not the Chekovs Gun it is in the finished film. It was added during reshoots when the filmmakers decided they needed an extra beat in the final battle.
Writing by the Seat of Your Pants: The script was not completely prepared when filming began, since the filmmakers were more focused on the story and the action, so the dialogue was mostly ad-libbed throughout filming and scenes were laid out to the actors in concept via outlines rather than scripted lines; Jon Favreau acknowledged this made the film feel more natural. Some scenes were shot with two cameras to capture lines improvised on the spot; Robert Downey Jr.. would ask for many takes of one scene since he wanted to try something new. Gwyneth Paltrow, on the other hand, had a difficult time trying to match Downey with a suitable line, as she never knew what he would say.