That infamous Nazgûl scream now sounds a bit like 4chan's REEEEEEE meme.
Galadriel's big speech about what would happen if she took control of the ring, since she's played by Cate Blanchett. Cate stars in Thor: Ragnarok as Hela - a death goddess who is indeed as "terrible as the dawn" and it takes a literal apocalypse to stop her.
Saruman's "So you have chosen death." experienced a surge in usage as a reaction quote in 2017.
Narm Charm: Boromir's anguished rant at no one, punctuated by falling into a pile of leaves. In any other film, utterly ridiculous. But Boromir's Tragic Hero status and Sean Bean's acting sell the hell out of it, and it's a near-Tear Jerker.
Figwit, aka "Frodo Is Great... Who Is That?", a random elf that got a sudden fanbase.
Sauron in his physical form in the prologue.
Signature Line: Gandalf's "YOOOOOOUU! SHAAAALL NOT! PAAAAAAAASS!" is arguably the most remembered and most often quoted line out of the trilogy. Even if he only said it a couple times in one scene in the movie.
Slow-Paced Beginning: Since the first act is spent entirely on exposition about the plot and contextualization about that world (Middle-Earth), it takes 50 minutes for Frodo and his friends to leave the Shire on their mission to destroy the Ring (in the extended cut).
When Aragorn and Frodo are on the Collapsing Stairs of Khazad-Dum, it is...rather obvious that they are in front of a green screen, with a fan blowing at their hair. (Understandably, since they couldn't possibly be filmed on a collapsing 500-ton staircase...) This was a very rare case of a failure that was quite easily visible on first viewing.
Not so much "special effects failure" as much as "director didn't catch it when filming" but in the scene where Aragorn runs to the dying Boromir (It's the next wide shot after he kills Lurtz, specifically) pay attention to the Uruk-Hai corpses. One of the extras raises his head to look around after Viggo Mortensen moves past him.
When the Fellowship is running from the Orc army in Moria, there is a scene just after their escape from Balin's Mausoleum where the tiny, running figures are clearly CGI characters rather than the actors themselves. If you look at them, rather than the Orcs gathering around them, you can see their legs aren't bending as they run, and their heads are swivelling evenly, as if they were all made of Lego.
Breather Level: The Prancing Pony stage of the Bree level, and the entirety of the Rivendell and Lothlorien levels (all of which come after levels that require you to fight through an entire gauntlet of enemies, along with at least one boss battle) have no combat, so Frodo can relax and talk with the other characters at his own pace before advancing through the story.
Ensemble Darkhorse: Tom Bombadil became this thanks to the fact the game has him from the book.
The fight in Balin's Tomb is abruptly skipped via Battle Discretion Shot in this game. Conveniently, EA's The Two Towers released shortly afterwards has an entire level based around it. And as a strange coincidence, Fellowship of the Ring ends with Aragorn and Frodo atop Amon Hen, with the latter being ordered to escape after the Winged Nazgul is defeated. The Amon Hen level in The Two Towers practically continues right where the Fellowship of the Ring game left off (despite being made by a different company), since it begins with Aragorn and Frodo atop Amon Hen, and Aragorn orders Frodo to escape while he and the others hold off the Uruk-hai.
That One Boss: The Balrog is an exceptionally difficult boss battle in the console and PC versions, especially since it constantly spams Gandalf with fireballs that will knock him down if they come into contact with him.