The entire movie could count as this to the franchise, as the "Search for God" plot is almost never referred to again after the fact.
Bizarro Episode: The movie is a textbook example; the plot ignores many of the rules and conventions of the setting, the entire premise comes out of nowhere, it has no impact on the ongoing plot of the films which kicks back in for the next one, and the events are never mentioned again. Removing it from continuity entirely would have no effect on anything else in the franchise. It's been noted as feeling a lot like Shatner wrote his own original sci-fi story, then simply changed the names to Star Trek characters.
"Funny Aneurysm" Moment: It's a bit uncomfortable in retrospect watching the scene in which Scotty hits his head on the bulkhead (after saying "I know this ship like the back of me hand!") knowing that James Doohan developed Alzheimer's disease towards the end of his life.
Kirk tells McCoy and Spock "I've always known I'll die alone." It was originally harsher after Star Trek: Generations, in which Kirk died without either one around; but it's heartbreaking now that William Shatner has outlived both DeForest Kelley and Leonard Nimoy. And then, in Star Trek Beyond, it's revealed that Spock has died, with the strong implication that he was the last surviving TOS crewmember.
When Sybok offers to take away his pain, Kirk refuses—emphatically insisting that he needs his pain. The next movie shows that Kirk holding onto his pain (specifically, his son being killed by Klingons) makes him an ample scapegoat for the conspiracy and is a major obstacle that he must overcome.
Sulu and Chekov have a few "old married couple" moments such as when they're lost in the woods in the opening. Which is even funnier after George Takei came out in real life years later.
Hypocritical Humour: One could view Gene Roddenberry's comments that he considered the film apocryphal to be this, considering that the Enterprise going on "the search for God" was his original pitch for the first movie.
Misblamed: Granted Shatner is the cause of a lot of the mess that is this movie, but he tends to get all of the blame even though there were several other factors such as Executive Meddling and the WGA strike.
When "God" chases Kirk near the end, it wails out "Yyyyyyyyyyooooooooooooooouuuuuuuuuuuuuuuuuuu!" It's supposed to be haunting and scary, but sounds like an elderly ghost from Scooby-Doo trying to scare someone, and failing miserably.
Kirk's speech about how he needs his pain, when you consider that it's actually Shatner writing a big speech for himself about how the central premise of his own story makes no sense.
The tagline "Why are they putting seatbelts in theaters this summer?" was meant to imply that the movie was so suspenseful you'd need a seatbelt to stay in your seat...or something. It was not meant to imply that the film was so boring that the seatbelts were to keep audience members from escaping, an interpretation many reviewers gleefully ran with.
Narm Charm: Sybok's final line before attacking the false God: "I couldn't help but notice your pain!...It runs deep! Share it with me!" Only he could take such a corny line and make it sound badass.
Never Live It Down: Sarek's disappointment in his "so human" son from the moment of his birth, despite Spock looking just as Vulcan as you could want.
Padding: A whole movie of it. You can easily skip from the fourth film to the sixth and lose nothing, and it actually makes the story flow better.
Nearly every damn special effect in the movie. The phaser and transporter effects (handled by the same team that produced the corresponding effects on Star Trek: The Next Generation) are decent, as is the Stock Footage of the Klingon Bird-of-Prey from the two previous films, but most of the effects would be barely passable for a movie made in the 1950s, never mind 1989.
Ironically, the Rock Monsters that were originally going to be chasing Kirk rather than the disembodied head of not-God were tossed out because the one suit they made looked "like crap," according to many. Seen here, we can see that they actually looked far better than the effects in the damned movie in comparison.
Most of the effects problems were apparently to do with the motion control photography being done at 16fps instead of the usual 24fps, as a cost-saving measure. Notably, the static shots of the Enterprise and the Klingon Bird-of-Prey are generally okay (if a bit flatly lit), but whenever they move they do it in a stuttery, jerky fashion that looks like something out of an old Ray Harryhausen flick.
On the topic of the Enterprise, the model representing it was vandalized during the film's production by employees of Universal Studios Florida. Basically, the fact that Bran Ferren (the man behind the effects) has never been allowed near another movie to this day (he currently works for Disney Imagineering) says it all.
Sha Ka Ree as seen from space is clearly a star, not a planet. On top of that, the planet's surface is clearly the same location used for Nimbus III, except tinted purple. Not surprisingly, this fails to produce the hoped-for effect of an ethereal paradise. Even the titular planet in the infamous TOS episode "The Search for Eden" looked far more like a tropical paradise than Sha-Ka-Ree does.
The Great Barrier is represented using an old-school FX tool called a cloud tank (essentially a large tank of water which other liquids can be injected into, creating surreal swirling patterns), but the results aren't particularly awe-inspiring, especially considering that the nebula from the second film was created using the exact same method 7 years earlier and looked far, far better.
The streak effect used when the Enterprise hits warp speed isn't bad per se but it's noticeably different than how the equivalent effect looks in the films handled by ILM, creating a bit of a visual continuity error. The transporter effect is much closer to ILM's but still a little less natural-looking.
Strangled by the Red String: When did Uhura ever show romantic interest in Scotty? Granted, it probably wouldn't have been out of place in TOS or one of the earlier films, but the attraction between the two comes out of absolutely nowhere, and is never referenced again.
The Nimbus III parts could have been amazing. Desert planet filled with rogues and criminals? It easily could have been Star Trek's Tatooine but alas the Shat happened. For comparison, Nimbus was revisited in Star Trek Online for a 5 episode arc which gave it a Fallout: New Vegas feel (a mix of Space Western and Steampunk) for a good story. Imagine if this movie had Kirk in a Space Western setting?
You'd think that with Sybok's powers to show people their deepest pains and release them, it'd be a good opportunity to show more of the backstory of the main characters, right? Unfortunately, that only ends up being the case for McCoy. Sybok doesn't get to use his powers on Kirk or Scotty at all, Spock's pain is something we already knew, and the Epiphany Therapy for Chekov, Sulu and Uhura happens off-screen.
Lawrence Luckinbill as Sybok, as seen on his official demo reel from the film. He gives a pretty decent performance and one suspects that he would have provided a truly memorable villain, if his character had been given a less ridiculous motivation and plot.
DeForest Kelley. Despite the film's generally hokey story and writing, a sizable contingent of fans consider this one of his best performances as Bones.