While Gene Roddenberry didn't like any of the movies, he really didn't like this one. Several of the cast and crew also have low opinions of the movie.
Though most of the cast and crew have admitted they enjoyed working with Shatner because of the level of energy he brought as director, others have been less kind. One of his unauthorized biographies (Captain Quirk) alleges he fired people for making fun of his toupee.
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-anniversary 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 States, The 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!" (due to the previous film's success in that area). This resulted in severe Mood Whiplash between a grand, epic story about the search for God and slapstick farce. Last, they 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.
The Merch: Along with the usual action figures and plastic models, Kraft offered a 'marshmelon' dispenser.
Old Shame: Virtually nobody involved with the film has anything good to say about it. William Shatner has since admitted that the film fell short of his vision for it and that it greatly harmed the franchise. Several cast members, among them George Takei also disliked the final product. The person probably the most positive about the film in retrospect is screenwriter David Loughery, and even he admitted that it could have been better had the writer's strike not cut into the time he had to work on the script.
Real-Life Relative: Bill Shatner's youngest daughter, Melanie, played the Enterprise yeoman.
Screwed by the Network: Retroactively. While most of the other TOS-era films were given "Directors Cut" reedits and re-releases, any of which allowed the directors to make improvements — albeit Star Trek III: The Search for Spock has never had any releases aside from its theatrical cut, whereas Star Trek IV: The Voyage Home only got an international cut that added a short pre-title sequence and a couple of other mild changes — William Shatner was famously denied the opportunity to put right this film with a Director's Cut, even though of all six original films, it was the one that needed fixing the most.
Wag the Director: A rare example where the actors arguably had better ideas than 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. Max von Sydow was the next choice, but he was too expensive.
The scene where Spock sees his inner pain was originally longer. Spock claims that he resolved any tensions with his father. He then has a flashback to his childhood: when he finds out Sybok has been banished from Vulcan, Spock says he wants to go with him.
William Shatner wanted Erik Van Lustbader to write the script. The author asked for $1 million and Paramount refused to pay up. Nicholas Meyer was then approached, but was unavailable, resulting in David Loughery eventually being hired.