Despite having a long history as a successful TV producer and being one of the two men — the other being Nicholas Meyer — that saved the Trek franchise with the second movie, producer Harve Bennett's career was utterly ruined in part by the failure of this movie. Afterwards, he wanted to do an "origin" story with the older characters as a framework, but Paramount wanted a big 25th-anniversery celebratory movie instead, and when after initially green-lighting his idea they then reneged, he basically gave up and quit the company. Unfortunately, this film's underwhelming box-office performance and critical drubbing made it virtually impossible for him to find employment elsewhere, and his only other major work afterwards was on Time Trax.
William Shatner's career ultimately survived The Final Frontier, although obviously any hope he had of becoming a major director went up in smoke afterwards.
The acting career of Cynthia Gouw (Caithlin Dar) didn't exactly take off either (fortunately she found work in other areas).
Effects artist Bran Ferren was at the beginning of a promising career, having previously done effects work for Altered StatesThe Untouchables and Little Shop of Horrors, but it was his work in this film (see Special Effects Failure) that torpedoed his career as an effects artist in movies. He ultimately ended up having a long career designing theme park attractions for Disney.
Executive Meddling: The original story idea from Shatner involved the Enterprise encountering Satan posing as God, and Kirk & Spock descending into Hell to rescue McCoy... which probably wouldn't have been suited to Star Trek anyway, really. The executives also demanded "More humor!" and slashed the film's budget, resulting in awful jokes and massive special effect failure throughout.
Franchise Killer: Very narrowly averted. The financial failure of this film resulted in the budget for Star Trek VI being so low it would have been impossible to produce a film, had Nicholas Meyer not called in a personal favor from the new head of Paramount and the studio wanting to commemorate the 25th anniversary of the franchise.
Wag the Director: Shatner's original draft included McCoy and Spock betraying Kirk along with the rest of the crew. However, DeForest Kelley and Leonard Nimoy flat-out refused, pointing out that it would be out of character for either of them, Spock especially, given how Kirk gave up everything to resurrect a Vulcan whose Famous Last Words included "I have been and always shall be your friend".
According to Shatner and several others in on the original creation process of the movie, the plot would have followed the current form of the film much the same... until they met God. Instead of being a random alien, this would turn out to be Satan and McCoy would sacrifice himself to spare Spock and Kirk — who would simply dive into Hell after their friend and drag him out of Hell with Satan nipping at their heels.
This was nixed by various other people working on the film as just being too polarizing and not fitting for Star Trek. Or something equally bizarre — which resulted in a string of compromises that resulted in a script far worse for the wear. Combined with the Special Effects Failure that would later fly up...
It should be noted that in one episode of TAS, they did go to the center of the galaxy and find Satan — though that story actually had the opposite denouement to Shatner's idea, and revealed that "Satan" was actually the sole nice guy in a world full of jerks.
The "rock monster" mentioned below under Special Effects Failure certainly also apply, at least in that it looked far better than the rest of the feature's effects. But as would be typical for this film's troubled production, it didn't work right on set and they couldn't do anything to fix it in post.
They had originally been hoping to get Sean Connery to play Sybok. The planet Sha Kah Ree was named so in reference to this.