The Danza: Barely counts as an example. Commander Thomas Griffin is played by Tom Wilkinson, but is only addressed once by his first name.
Enforced Method Acting: In 2, when the stars were hanging from the bamboo wall, Jackie Chan decided to play a prank on the acrophobic Chris Tucker and told him that the stunt people forgot to attach his safety harness. When the already terrified Chris literally started crying, Jackie realized he had gone too far.
Fatal Method Acting: Averted. A stunt nearly crushed Jackie Chan's skull between a pair of metal boxes. They slammed together about a quarter of a second after Jackie's head was clear. If he were any slower, he'd have been dead.
Irony as She Is Cast: In the third film, Lee tells Carter that he doesn't speak French, which is untrue for Jackie Chan in real life.
Money, Dear Boy: Jackie Chan kept starring in the films because he was offered an "irresistible" paycheck to continue making them. That said, by most accounts, he did enjoy making them and built a close rapport with Chris Tucker whilst filming.
Promoted Fanboy: Roman Polański was a fan of the first two movies. When it was learned that Rush Hour 3 was set in Paris, he asked Brett Ratner and the producers whether they could give him a small role to appear. They happily complied with his wish.
Throw It In!: Chris Tucker improvised much of his dialogue, as he normally does in his films. According to Brett Ratner, during the scene at Grauman's where Detective Carter bribes Stucky for information, there was so much improvisation between Tucker and John Hawkes that they almost did not think they could edit it together as a coherent conversation. There are still continuity errors in the dialogue for this reason.
Translation Convention: The series pointedly averts this, featuring a great deal of Cantonese, and in the third film's case, French.
Troubled Production: The production of the second film went pretty smoothly, but it has become infamous for a single incident that occurred as it was wrapping. One scene involving a bust on a counterfeit ring in Las Vegas was to end with a pile of money going up in flames and $100 million worth of bills floating through the air. The scene was shot without issue, but the prop bills they had used were so convincing that extras and bystanders had picked them up and attempted to use it as actual legal tender on the Vegas Strip and beyond. The Secret Service ended up intervening, accusing the prop maker, Independent Studio Services, of counterfeiting, and forced them to recall and destroy all the money they had sent out to other productions. ISS did manage to survive the ordeal, but not without significant cost. (A rather ironic case of real life imitating art; the film itself centers on the protagonists trying to stop a massive money-laundering scheme, using counterfeit bills closely resembling U.S. legal tender—assisted by a Secret Service agent, no less.)