Crack Pairing: Uhura and Scotty? Seriously? 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.
Fridge Brilliance: A possible reason for the numerous problems the almost mint-condition Enterprise-A is having in this movie? It was in Spacedock during the Probe's sojourn to Earth.
Alternatively, Roddenberry suggested it's problems were because it was actually the Yorktown, having been renamed and rushed back into service after the loss of the Enterprise, hence why it comes across as a second-hand ship.
If Scotty knows the ship "like the back of his hand", why does he end up smacking his head on an overhanging bulkhead? Because he was likely thinking of the Enterprise, not the Enterprise-A. Furthermore, it's likely that not all Constitution-class vessels were refit in exactly the same way, particularly if this ship was originally the Yorktown.
It's never outright stated, but as noted below and elsewhere, it's a safe bet that Kirk's greatest pain is the death of his son.
SF Debris has pointed out that the movie becomes a lot better if you view it as a satire of Roddenberry's ideal "Communist sex utopia" future.
Obviously, for whatever reason, once it's flushed and resealed the waste is vented right into space. Which, when in spacedock, would mean probably splattering some poor maintenance worker's windshield.
The same reason you can't use train toilets in the station. Anywhere else on the Enterprise they probably just use the transporter...
"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.
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.
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. Though for a good contingent of fans it's just one more piece of crappy writing better left forgotten.
Not to mention it's just bone-headed for Sarek to be disappointed his son is "so human" considering SPOCK'S MOTHER IS HUMAN. AND HE KNEW THIS. Freaking duh he's human!
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.
Retroactive Recognition: By far the best remembered acting role by Lawrence Luckinbill, who's now far better known for being the uncle of the Wachowskis.
Critic: At times, it can have some good moments, even some good character development..."
Special Effect Failure: 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.
They look pretty decent standing posed for still photography, but that's still different than looking good in motion and intended to be watched on the big screen. Also there were apparently worries about the wearers' safety, since the suits were designed to emit smoke but the device to do so kept malfunctioning.
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.
Took the Bad Film Seriously: Lawrence Luckinbill as Sybok. 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.
Also 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.