Same can be said for Shatner, but people tend to ignore that one.
Shatner was, according to "Star Trek Lives," scheduled to direct the final episode of the Original Series in 1969. Unfortunately, that would have been Episode 81, and NBC sent the word down during the filming of "Turnabout Intruder" that that episode, #79, would be the last.
In his book I Am Spock, Nimoy mentions that DeForest Kelley kept swearing up and down that, in the scenes where McCoy converses with an unconscious Spock, Nimoy was trying to direct him through the movement and flutter of his eyelids.
Fake Russian: In letter, but not in spirit. Chekov actually speaks Russian for the first time. Though Walter Koenig was born in America, his parents are Russian immigrants who emigrated from Lithuania.
The Other Darrin: Kirstie Alley loved playing Saavik (she apparently wore the ears home a number of times) and wanted to return, but apparently failed to communicate this clearly to her agent, who demanded a higher fee for her to play Saavik than Paramount was willing to pay. Paramount switched in Robin Curtis instead.
To be quite exact: Shatner stumbled on a previous take, whereupon the two convened and decided that the scene should include such a gesture. The recorded take was then staged, leading to the movie we now know.
The official Klingon language was forced to include some complex structural rules entirely because Christopher Lloyd screwed up one of his lines and it got left in the film. However, Michael Okuda realized that the mistake Lloyd made actually made sense because it was basically a contraction of what would have been excessively formal if spoken in English.
Early scripts have the Klingons as a rogue group who commandeered a Romulan Bird of Prey.
And earlier than that, it was supposed to be Romulans, period, in Harve Bennett's treatment. Spock's role is relatively the same, except he goes feral and kills several Romulan miners when they drill for dilithium on the Genesis planet. A subplot would have also had the Vulcans threatening to secede from the Federation over the Genesis Torpedo... The 2009 reboot would have temporally-displaced Romulan miners as the Big Bad.
In the novelization, Saavik and David Marcus became romantically involved (the seeds for this were planted in the previous movie's novelization); this storyline was completely dropped from the films.
In the novelization, Saavik became pregnant with Spock's kid. This, too, was dropped.
There was some debate as to who would get killed off at the end, Saavik or David Marcus. Ultimately they chose to have David make a Heroic Sacrifice to atone for the damage he had done with the Genesis Project and also to balance out the return of Spock.
Gene Roddenberry would have preferred it had the Enterprise just lost its saucer and not the entire thing, envisioning the movie ending with a new saucer being attached to the old secondary hull, making it a fusion of new and old and giving it a sense of legacy.