"The Warlock's Hairy Heart" — A young rich warlock fancies himself Genre Savvy to the fact that Love Hurts and cuts out his heart to save himself the hassle. Then he finds out that everyone thinks he's a loser for not having a girl and sets out to bag himself a trophy wife. This does not end well.
"Babbitty Rabbitty and her Cackling Stump" — A Muggle king decides to eliminate wizards and learn magic himself, unaware that it does not work that way. He asks for an instructor in magic and only a fraud responds since real wizards were not born yesterday. Then the fraud encounters a real witch...
Anvilicious: Invoked. The Tales of Beedle The Bard were essentially the wizarding equivalents of Aesop's Fables.
Asshole Victim: Loxias, a wizard who at one point had the Elder Wand, according to Dumbledore's notes on The Three Brothers. Several people, including his own mother, claimed to have killed him.
Beat Still, My Heart: You can guess from the title of "The Warlock's Hairy Heart" what happens when the warlock in question removes his heart from its usual place in an attempt to keep himself from falling in love.
Disneyfication: Spoofed: In one of Dumbledore's commentaries he describes how a Beatrix Bloxam republished Beedle's tales, taking them to ridiculous heights of Tasting Like Diabetes - her wizarding card in one of the video games even says her book was banned for inducing vomiting. He later explains that this happened after she was horrified by "The Warlock's Hairy Heart", which incidentally she was never able to "sweeten" to her satisfaction. All this, and the fact that kids hated the Bloxam versions, might very well be a Take That against the many real-life Moral Guardians who worried that Harry Potter was traumatizing children.
Draco in Leather Pants: In-universe. After all her good feelings from her times with her lover are gone, Amata realizes he was actually a Jerkass and is glad to be rid of him.
Fair for Its Day: "The Wizard and the Hopping Pot" and "The Fountain of Fair Fortune" were controversial in-universe, especially at the time they were published, for their positive depictions of wizards helping and marrying Muggles, respectively.
Fighter, Mage, Thief: The three brothers, or more precisely Death's gifts to them, fall under each of the three archetypes: The Elder Wand, being unbeatable and giving its owner lots of raw power, falls under the fighter archetype; the Resurrection Stone, being preternatural even by wizarding standards, falls under (the wizard equivalent of) the mage archetype; the Invisibility Cloak, naturally, falls under the thief archetype.
Footnote Fever: Thankfully they're all compiled into their own mini-chapters between each Tale.
Freudian Excuse: Beatrix Bloxam's hatred of anything not overflowing with Glurge apparently stems from her overhearing "The Wizard's Hairy Heart" as a child and being traumatized by it. That it somehow segued into "ghastly details of the dreadfully unsavory affair of my uncle Nobby, the local hag and a sack of Bouncing Bulbs" didn't help, she was in bed for a week after that.
Friendly Enemy: "The Tale of the Three Brothers" ends with the youngest brother voluntarily dying after evading Death with the cloak of invisibility after years and years. They had apparently arrived at something resembling friendship at that point.
The Grim Reaper: More simply known as Death in "The Tale of the Three Brothers".
In Harry Potter and the Deathly Hallows, Hermione receives a very old copy of The Tales of Beedle the Bard, a traditional book of wizarding fairy-tales. This book appears as a translation from the original runes made by Hermione, accompanied by a set of notes on the tales written by Albus Dumbledore, with J. K. Rowling only adding a foreword and some notes for us Muggles' benefit.
It mentions "the seven volumes of Harry Potter's biography", thus making the HP books real in their fictional universe.
Maligned Mixed Marriage: At the end of "The Fountain of Fair Fortune". As indicated by his letter, Lucius Malfoy did not approve.
Misaimed Fandom: In-universe. "The Tale of the Three Brothers" is a story about the futility of beating Death, but many wizards who heard the story figured they could do just that if that had all three Hallows.
Dumbledore's bemoaning over the continued folly of wizards to seek to defeat death and his observation that "humans have a knack for choosing precisely the things that are worst for them" all take a serious tint of Heroic Self-Deprecation when you remember that he's guilty of every single one of those things, and knows it all too well.
Voldemort takes it a step further: having never heard "The Tale of the Three Brothers" in the first place due to his Muggle upbringing, he has no idea of the true significance of the Elder Wand (or that it's part of a set) - he just wants it because he believes it will make him invincible.
Moral Guardians: Besides Beatrix Bloxam's revisions mentioned above, Dumbledore notes that some tales got rewrites through the ages because they were deemed too pro-Muggle. Specifically, he explains that the Hopping Pot went from causing trouble for the mean wizard who refused to help his Muggle neighbours, to swallowing threatening Muggles until they left the nice wizard alone.
Also, Lucius Malfoy once tried to have The Fountain of Fair Fortune censored, because the witch Amata marries the Muggle Sir Luckless at the end of it. Dumbledore, however, gave a thorough speech about keeping it in the library by saying that almost every witch and wizard are at least half-blood.
Nightmare Fuel: "The Warlock's Hairy Heart" is an In-Universe example (Dumbledore points out many parents only read it when the children are grown).
Riding into the Sunset: The wizard and the hopping pot do this at the happy ending of their story, when the wizard learns his lesson.
Romance on the Set: In the commentary for The Fountain of Fair Fortune, Dumbledore mentions the unfortunate casting choices in a theatrical version of the aforementioned story - the students playing "Amata" and "Sir Luckless" had been dating until "one hour before the curtain rose," at which point "Sir Luckless" dumped "Amata" for the girl who was playing "Asha."
School Play: In one of his commentaries, Dumbledore describes how Hogwarts once had a disastrous production of "The Fountain of Fair Fortune." The disastrous results prompted a ban on all theatrical productions at Hogwarts.
Second Love: Sir Luckless is this to Amata at the end of "The Fountain of Fair Fortune".
Single Woman Seeks Good Man: The ending of "The Fountain of Fair Fortune" - the third witch, Amata, whose heart had been broken, moves past her disastrous first love and comes to love Sir Luckless.
Take That: In-Universe, at the end of the commentary for the Fountain of Fair Fortune: "This exchange marked the beginning of Mr. Malfoy's long campaign to have me removed from my post as headmaster of Hogwarts, and mine to have him removed from his position as Lord Voldemort'sFavoriteDeath Eater."
The whole last page and a half of the commentary would suffice, including the last footnote.
Plus, in the commentary on "The Tale of the Three Brothers", Dumbledore delivers one to those scholars who make up BS about past authors and their works for their Piled High and Deep.
Terrible Ticking: The Hopping Pot for the wizard who inherits it from his father, who used it to help their Muggle neighbors before he died. When the wizard denies the Muggles' requests for help, the pot starts hopping and making noises that are reminiscent of the suffering the Muggles endure as a result of the wizard's selfishness. Eventually, this drives the wizard to relent and thereafter start following his father's example.
Unreliable Narrator: Dumbledore's notes on the Tale of the Three Brothers contradict or omit some things we know he knew about the Hallows. This is lampshaded by the very Prologue of the book, which more or less tells you to make your own opinion as to why Dumbledore played ignorant.
Women Are Wiser: "No witch in history has ever claimed to own the Elder Wand. Make of that what you will." The implication being, of course, that women are too smart either to want the wand or to advertise that they have it.