I'm not sure exactly what you're hoping for here. The Aarne-Thompson Tale Type Index does not include "suicide" in the subject index at the back, and I can't recall any folktale that specifically deals with this. I assume by "fairytale" you mean folktale - the folk's novels: long-multi-episodic stories that are not to be believed. Many cultures have legends (shorter stories that are told to be believed) about the ramifications of suicides. These usually involve restless ghosts or the failures of spirits to enter heaven. I do not recall any supernatural beings of nature (which some culture call "fairies") as being associated with these stories.

Stith Thompson's motif index (motifs are the bits that are combined to make up folktales) includes an entire page of references to suicide in his subject index (volume 6). The references are diverse, but I am not seeing anything that specifically references supernatural beings other than ghosts. If you can give me more direction, I may be able to help.

Greek mythology might have some ideas for you.

The Greek concept of the furies, or the erinyes, might correspond to your idea of a fairy… they were tormenting gods of vengeance that harrassed, notably, Orestes (who killed his mother, see Aeschylus' "Oresteia" or Euripides' "Orestes" plays), and maybe also Oedipus (if you read Sophocles' "Oedipus at Colonus"), who - as you know - unknowingly killed his father and married his mother. Be warned that the furies weren't associated with suicide, but with serious moral crimes like murder, or anything that offended human nature.

You might want to look at Euripides' "Iphigenia in Aulis" which is a play in which the titular character, the daughter of Agamemnon, is supposed to be killed as a sacrifice to the gods. After initially resisting the prophecy that condemns her, she eventually consents. At the last moment, a deer is sacrificed in her place.

Or maybe you could look at the myth of Narcissus, who fell in love with his own reflection and dies contemplating his own beauty in a pool of water.

Outside of Greek myths, you could also look at the "Song of Roland," which is an early French epic poem. The brave warrior Roland and his men are defeated at the battle of Ronceveaux, but before all is lost, Roland blows his olifant (sort of like a trumpet) to inform Charlemagne, his king, of his defeat. He blows the horn so hard he dies and is taken by angles to heaven. This story also inspired two Italian adaptations, Rolando innamorato and Roland furioso.

Let me know if you want more information, I'd be happy to write more. You also might want to try at /r/askliterarystudies …

Not sure if this deserves a top post, but when you mention fairies The only ones I would know of are the Tuatha de Danaan. As far as I know, while there are many fairies who visit prior to death (one of the more famous accounts is Cuchlain seeing his bloodied clothes being washed while on his way to battle; the Morrigan would also appear to some prior to their death), I'm not certain of one specifically associated with suicide (though it has been a while since I've read the myths, only suicide I can think of at the moment is Diedre with the Red Branch). These general death fairies may start you off to where you want to go, when I get back home I'll go through some books.

Just as a word of caution, my source text tends to be T.W. Rolleston's Myths and Legends of the Celtic Race. It's a bit outdated at this point, so many views have changed, although the myths themselves tend to remain consistent.