A Crown of Stars: In chapter 49 a character is being filled in on the history of the Angel War and the post-Impact world. However Misato had some creative interpretations of the events. Asuka suggests him that he ignores everything Misato said.
Elspeth of Luminosity narrates the second book, and whenever under Allirea's power sees her as "not important". This leads to glossing over some important dialogue, with a little Unspoken Plan Guarantee.
The museum curator from "The Courier Who Had Cheated Death" averts this trope. On one hand, every detail from the story he told was true. On the other hand, he was the murderous psychopath from the story, and the 'display dummy' he mentions offhandedly is implied to be another of his victims.
Hunting The Unicorn makes liberal use of this—though there isn't any intentional misleading, there are two instances that make use of this for huge impact: "The Hunters" reveals that Blaine isn't a virgin, and it's elaborated very painfully in the following chapter. "The Butterfly" is where David tells a counselor that Blaine has a stalker and has no idea of it.
In the Pokémon fanfic Revenge of the Narrator, the replacement Narrator tells the reader halfway through that everything the original Narrator had said was a lie.
Pipeline is primarily told through third-person limited, using Kevin's thoughts and perceptions of things to tell the story. Kevin's kind of a... self-informed guy, so this has interesting results. He's got the best of intentions, really, but his perceptions of the way Ben is acting towards him are much harsher than Ben means them to, and his irrational dislike of Dexter makes the boy genius out to be the bad guy sometimes when they really have similar values and goals.
In the Batman fanfic A Piece Of Glass, the story is sandwiched together from the POV of the Joker, his Original Character accomplice, Breech Loader, and Batman himself. The Joker sees his demented social experiments as perfectly acceptable. Breech repeatedly insists that morals and sanity are moot points, being a matter of perspective. Neither is sane, but Through the Eyes of Madness both are convinced they are right. Interestingly, through Batman's POV, he's Not So Different...
Similarly, Dogbertcarroll's stories on Fanfiction dot net include one where Xander Harris has a cosmic event happen and gets dropped into the Justice league, literally during a meeting. He describes Batman as being possibly the most sociologically driven man in the DC Universe but also deadly necessary. Of course, this is Xander's viewpoint, and the man has been somewhat unreliable narrating himself, as anyone who's ever watched/critiqued "The Zeppo" can tell you.
All over the place in The New Retcons since for most of the story it's written in the style of the characters writing letters. Including an insane Elly. In the comments for one of the letters, the authors and fans discussed this trope with regard to Liz, and whether she was one about the going after.
Possibly Dominic in Pink Personal Hell And Altering Fate. Early in the story, he tells the reader how bad he has it, such as how his group doesn't bother to communicate with him and dumps half the project on him. However, when he arrives to give a presentation with Gummy attached to his finger (It Makes Sense in Context) they actually laugh with him. Granted, later on, the mirror shows moments he'd rather not see...
One thing that also makes a bit of sense with the story's main twist is Pink Personal Hell is revealed to be In Medias Res to Altering Fate - it can be read in a way that Dominic is remembering his so-called "Pink Personal Hell" during the events of the "Altering Fate" narrative from his perspective - which still plays true to this trope as Dominic glosses over a lot of events.
Home with the Fairies downplays this. Maddie is not trying to lie, but her misunderstandings affect the narration, especially in the early chapters, when the Language Barrier is still a major problem. For example, Maddie visits the town of Fornost, but it might not be Fornost; Maddie later uses the name "maybe-not-Fornost". Then in chapter 13, Maddie believes that Lord Kinsey will fire her if "gossip gets out", but this might not be true; Lord Kinsey might or might not believe the gossip. A writer's note on chapter 14 declares Maddie as an unreliable narrator.
The Pokémon fic Obsession shows Corbin as a caring father who simply doesn't know how to care for his strange son. Said son is narrating, however, and describes his father as a heartless fool and constant embarrassment. He's also narrating as an adult, so this isn't just a child's perspective.
Brett's mother in The Legend of Total Drama Island. Although she recounts long-past events with inhuman precision, she also embellishes some details, fills in gaps with informed guesswork, and lets her biases influence some characterizations. Indeed, those cartoonish elements in the original that are retained in the reimagining could well be chalked up to her embellishments. It’s called a “legend” for a reason.
For example, he believes that Miku Hatsune was the first Vocaloid preceding Meiko Sakine and Kaito. He's convinced that Kino from Kino's Journey is a dude.
It should also be noted that several context clues within the same chapter(s) strongly suggest that this is not a critical research failure on the actual author's end.
The narrator of Equestria: A History Revealed is without a doubt, one of the most unreliable narrators to ever be featured in a fanfiction. Her tendency to present her conspiracies as fact is both disorienting and highly amusing as well. But it is this nature of hers that the entire concept of the fic centers around.
It is possible to get a glimpse of actual Equestrian history through her eyes, once one wades through the enormous fallacies and insane conspiracy theories she presents. But the fic mostly consists solely of Equestrian history as seen through her eyes, whether the reader wants to accept it as accurate or not.
Navarone is an in-universe example in Diaries of a Madman, as he often leaves stuff out or puts misleading information in his journals. Discord is a straighter example, as he flat out lies to the reader.
All over the place in I'm Here to Help. Emerald's narration portrays Crystal Tokyo as a Crapsaccherine World where everyone is brainwashed and the senshi rule with an iron fist, but the only reasons actually given for how the kingdom is bad boil down to "it's boring". It doesn't help that Emerald is several hundred years old and insane. (Pluto's interference near the end could indicate some truth to his argument, but she gives no reason for her actions beyond "I don't like how Crystal Tokyo turned out", which still gives nothing solid to go off of.) Meanwhile, the two sections told from the point of views of the future senshi depict Emerald as a dangerous murderer, while the past senshi and Luna see him as shifty and untrustworthy. It's difficult to say exactly how much of this was intentional. The author's notes at the end say that there was a lot more going on than we see, but we're never told what it was. The possibility of everything being revealed in a sequel was mentioned, but that never came around.
Carlos in his chapters of ''Compliance'' and ''Procedure''. His narration includes, for instance, domestic abuse, emotional manipulation not being mentioned as such, and sexual assault/rape being seen as consensual sex (the following chapter proves that it very much is not).
The Sassgardian is the worst offender in Superhero RPF. He insists for example that Loki is a Lovable Rogue with a heart of gold, he even has his tumblr to prove it! He is Loki. But several other characters are super heroes or villains too, protecting their secret identities, so being unreliable narrators is pretty much unavoidable.