In mothy's Moonlit Bear, Eve Moonlit, the character Vocaloid Hatsune Miku plays in the song, talked about how she found two apples deep in the wood and got chased by a bear. As it turned out, the apples are two infants and the bear their mother, whom Eve ended up killing.
Most of The Barenaked Ladies song "The Old Apartment" is meant to imply that the narrator has broken into his ex-girlfriend's apartment in a fit of creepy stalkerishness. Toward the end of the song, he reveals that he and the girlfriend are still together, and have just moved to a nicer house; he's broken into their old place in a fit of creepy nostalgia.
The protagonist of King Diamond's concept album The Graveyard claims that he was thrown into a mental hospital because he threatened to expose a politician as a child molester. Since the entire album is from his point of view, and he's an insane killer, it's not clear if he's telling the truth or just crazy.
The refrain of Gaelic Storm's "Johnny Tarr" goes: "Even if you saw it yourself you wouldn't believe it/But I wouldn't trust a person like me if I were you/Sure I wasn't there - I swear I have an alibi/I heard it from a man who knows a fella who swears it's true". The story told in the song is borderline fantasy, wherein the title character dies of thirst in the middle of a drinking contest.
They Might Be Giants do this so much they considered calling one of their albums Unreliable Narrator. To cite one example, "Purple Toupee" is built around the narrator's horribly mangled memories of newsworthy events of the 60s ("I remember the book depository where they crowned the king of Cuba"..."Martin X was mad when they outlawed bell bottoms").
Denton, TX based Slobberbone's "Billy Pritchard" features a father telling his daughter how he doesn't approve of her relationship with a boy in her town, and implies that he killed her brother. Near the end of the song, we learn that the father shot his own son in the back of the head after mistaking him for Billy, and that most of what he had said was a lie.
Eminem played with this for the majority of his career. His 'Slim Shady' character was an obvious parody of the excesses of the gangsta rapper archetype, but a lot of the devices Eminem used with Slim Shady were kept on even after he abandoned the character. How much of Eminem's rapping reflected his own attitudes is a very debatable question.
In Joanna Newsom's song "Colleen", the story is told by a young mermaid or sea nymph who lost her memory and was subsequently adopted by humans. It's implied that by the end of the song, she's still unaware that she's not human, although it's obvious from the lyrics.
I'll tell it as I best know how, and that's the way it was told to me: I must have once been a thief or a whore, then surely was thrown overboard, where, they say, I came this way from the deep blue sea...
Ludo's song "Lake Pontchartrain" is told from the perspective of a young man who supposedly witnessed his friends' watery, supernatural deaths. But at the last verse he adds; "That's how it happened/Why would I lie?/There were no bodies/I got none to hide", implying that he's being tried for killing them.
Gorillaz bassist Murdoc is notorious for this. He may be the only speaking witness to Noodle's disappearance and apparent death, but he changes the story every time he tells it. Sort of justified in that he claims to be withholding information in hopes of a movie deal. Of course, Murdoc's been known for exaggerating stories and flat out lying on important topics, so it's possible that he's just making things up as he goes.
Regal Pinion's songs has some of this. Sometimes the narrator's don't know if they can even trust themselves.
In Napoleon XIV's "They're Coming to take me away HAHA," the main character went mad because his wife divorced him (or his dog ran away, depends on the interpretation). In another song in the series sung from the point of view of the wife\dog, it shows he wasn't really all there beforehand.
The Nick Cave song "The Curse of Millhaven" from the album Murder Ballads introduces us to Lottie, a young teen girl who recounts the nasty murders that have been plaguing her small town. By the end of the song, it's revealed that Lottie herself is the curse of Millhaven and has been committing all the killings.
The Silverstein song "A Great Fire" and what follows throughout the album "A Shipwreck in The Sand". The first song talks about how the protagonist/hero saves his wife and daughter from their house that's burning down.though there are some things in the song to hint at something not quite right with how the husband wife treat each other;this was my home/this was my life/it's not always just about you. it doesn't become apparent till a later song what happened to cause the fire. in "I Am The Arsonist" he set the house on fire himself, because in the second song "Vices" he found out his wife was cheating on him, which lead to drinking, trying to hide he knew and knew how disappointed in him his wife was. it culminates in a song just before the last 2, "A Hero Loses Everyday", in which he states; The Protagonist became/The villian they disdain/In every way and ends on a realization that they could never have truly loved each other in the first place, because they were broken people.
In the Mercedes Lackey / Frank Hayes song The Leslac Version, Leslac the bard tries to tell the story of wandering heroes Tarma and Kethry liberating Viden town, but Tarma keeps interrupting with snarky corrections. In his version they deliberately sought out the tyrant to bring him down; in her version he died accidentally in a drunken bar fight. He plays up their nobility, she plays it down, but the truth is probably closer to Tarma's version:
Leslac: They searched through all the town to find and bring him to defeat.
Tarma: Like hell what we were searching for was wine and bread and meat!
Leslac: They found him in the tavern and they challenged him to fight.
Tarma: We found him holding up the bar drunk as a pig that night!
Lackey went on to write a short story about the events surrounding this song. Tarma's version has a few minor inaccuracies, but Leslac's version is complete nonsense. The amusing thing is that Leslac was present for the events of the song, but ultimately decided that he couldn't write a song about how a belligerent drunk (Who coincidentally happened to be the unpopular local lord) picked a fight with a couple travelers for no intelligent reason, got hit with a broomstick, and accidentally broke his skull against the fireplace and died. So he wrote a song about how the story should have gone. In fact, the author invented Leslac to handwave away mistakes she wrote in some of the Tarma and Kethry stories due to the fact that she wrote some of the songs about them before the stories behind the songs, and forgot a few details. All mistakes in the songs are Leslac's either because he didn't do the research, or changed the story to be more dramatic.
Bee Gees: "And somehow in this madness believe she was mine -but...I'm a liar"
In Tom Waits' "Christmas Card from a Hooker in Minneapolis" a woman tells an old acquaintance named Charlie (possibly an ex) that she's cleaned up, got married, has a child on the way (though it isn't her husband's) and is happy for the first time since an unspecified accident. She then admits it's all a lie; in fact she's lonely, in debt and is writing to Charlie to ask for money. She concludes by telling him she'll be "eligible for parole on Valentine's Day".