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Beam Me Up, Scotty!: It's often claimed that there isn't a single weapon fired in the movie. This isn't really correct, as there are two actual weapons firings (Kirk using his phaser to weld a door shut, and the whalers trying to harpoon George and Gracie) and one attempted (Chekov trying to stun the interrogation officers). This is likely a mix-up with a real fact, namely that this is the only Star Trek film with a body count of zero (assuming everyone survived the Probe's "attack", though Word of God is they did).
Reality Subtext: At the end of the movie, when the crew are speculating what ship they're going to get:
Sulu: I'm counting on Excelsior.
Scotty:Excelsior? Why in God's name would you want that bucket of bolts?
This is a twofer: Harve Bennett, producer and writer of the film, had initially wanted the crew to end up commanding the USS Excelsior (which had been seen in the previous film) but was overruled; and Sulu himself ends up as captain of the Excelsiortwo films later.
Technology Marches On: Gillian mistakes Kirk's communicator for a pocket pager. Also the humour provided by Scotty using the then-cutting-edge Macintosh Plus as if it were hopelessly primitive is ironic nowadays.
The Other Darrin: The USS Enterprise that Chekhov and Uhura sneak into is actually USS Ranger, a then-standard oil-fired carrier.
They actually had Nichols and Koenig asking real San Franciscans where the nuclear submarine in Alameda was. The police officers staring suspiciously at the pair was a real cop who didn't know what was going on. The woman who answers, "Ooh, I don't know. I think they're in Alameda", had to be chased down to sign a release form for providing such a hysterical line in response to their queries. See here for details.
Leonard Nimoy was trying to crawl out of the water during the final scene, but the rest of the cast members just dragged him back in.
The scene where Spock nerve pinched the punk rocker originally ended with Spock turning the punk's radio off manually, but they went with the take where the radio shuts off from the punk's head falling on the button.
What Could Have Been: Eddie Murphy was originally going to play a conspiracy nut radio personality that filled the story role of the marine biologist Kirk interacts with (minus the romance, obviously). He eventually backed out of the movie but it resulted in a massive change to the storyline. The producers were worried that if Murphy was in the movie it would turn into a guest star movie like Superman III became Richard Pryor's movie. So they made sure the biologist character was clearly a secondary character to Kirk and Spock.
There was supposed to be a scene where Sulu encountered a little boy whom he would realize was actually his ancestor, but when the time came to film it the child actor who had been hired became upset and could not be made to act properly, and unable to find another child actor, the scene was cut.
There was supposed to be a scene of Sulu stealing the helicopter, but it required him to jump into the chopper and George Takei couldn't do it because he was too stiff legged from running in the San Francisco marathon. They tried filming it with a stunt man throwing him inside but that didn't look right so it was cut.
Saavik was supposed to be revealed to be pregnant with Spock's child from the scene in the previous film where she had to, ahem, help him through puberty, which was to explain why she didn't go back to Earth with the rest of the crew to serve as a witness on their behalf at the trial, but it was cut for unknown reasons.