Older Than Feudalism: Homer did this in The Iliad. He went to the trouble of describing the battle between Aeneas (mortal son of Aphrodite) and Achilles. The Gods saw that Aeneas was about to be killed, and, since he was such a good servant, decided that he didn't deserve to be killed, and took him away from the battle, declaring that he would be the future king of all Trojans yet to come. Homer never mentions him again, and even the other authors of the lost epics only said that he either fled Troy after a bad omen or was captured and spared by the Achaeans. Later Greek authors said that he went to Italy. It took eight hundred years for Virgil to turn this into a Brick Joke. An EpicBrick Joke, at that!
In the Vitae Sanctorum Britanniae, or "Lives of the Saints of Britain", which predates the 9th century, warfare breaks out when a lovely maiden is kidnapped by King Maelgwn's soldiers. St. Cadog approaches the king and convinces him to repent and recall his army - but never asks for the maiden back, despite her father being an official in Cadog's church. She is never mentioned again.
When the narrator of Edgar Allan Poe's "The Fall of the House of Usher" arrives to visit his childhood friend, he's greeted by a footman and valet in Roderick Usher's employ. There are no other mentions of household servants in the story, and no hints as to whether or not any of the staff live in the House and are killed when it suddenly collapses at the end.
In The Phantom Tollbooth, Faintly Macabre, King Azaz's aunt, is imprisoned for having once abused her position as "Offical Which". She tells Milo the history of the kingdom, noting that she would only be freed once the Princesses Rhyme and Reason were restored from exile. Yet, we never see Faintly Macabre again at the end celebrating the princess' return (nor the Whether Man either, for that matter). Hopefully, she was let out soon thereafter.
Some readers would also like to know who changed the court docket to interfere in the Wakecliff inheritance case and why. Finding out which claimant won would also be of interest.
In George Orwell's Animal Farm, the cat (who never did anything of import while he was present) ceases to exist about halfway through the story. Orwell just stops mentioning him. This could be intentional, as the cat was already mentioned to be mysteriously missing every time the animals needed to get work done, and disappears right when the pigs are beginning to institute their crazier forms of tyranny.
In Robert Bloch's The Yougoslaves (sic), a gang of murderous, brainwashed boys is shown raping a little girl. The boys are eventually killed. No mention is made of what happens to the girl.
This was recently resolved with Wraith Squadron due to the book Mercy Kill, but earlier the fate of most of the members of the squadron and what they had been doing for the past couple of decades was left up in the air.
The Black Fleet Crisis trilogy ended with formerly enslaved Imperials stealing their fleet back from the Yevetha and heading for the Empire's backup capital, Byss. They then completely disappeared from the canon, a state of affairs only rectified in The New Essential Chronology which stated that the Black Fleet ended up at Byss to discover it had been obliterated by the Galaxy Gun. The fleet fractured, with some ships defecting to the New Republic, some heading for the Imperial Remnant, and SSD Intimidator disappearing. The New Republic ran across its wreck a few years later.
The Green Mile, there's a literal "What Happened to the Mouse?" when Mr. Jingles runs away after Eduard Delacroix is executed. Stephen King wrote in the afterward that even he forgot about Mr. Jingles until his wife asked him the question, so he wrote in a resolution. The mouse lived to the age of 60 years - twenty times the normal lifespan of a mouse.
In his short story "The Jaunt", the protagonist's daughter almost literally asks this question when the protagonist tells the family the story of the eponymous teleportation device's invention. To wit, the inventor ran down to the pet store and tested some white mice out on the machine. Slightly subverted when he euphemistically explains that they "didn't feel so good the first time" after being sent through awake. And by now you've probably guessed why he was being euphemistic with his family about what happened to those mice.
At the end of the flashback section of Wizard and Glass, Roland's mother gives him a belt. He promises to tell his ka-tet the story of how he lost the belt, "for it bears on my quest for the Tower." Whether he tells the ka-tet or not, he never tells the reader; the belt is never mentioned again either in the main series or in any of the side materials.
At the end of Frankenstein, the doctor mopes that everyone he loves is now dead. However, Shelley never mentions what happened to his brother Ernest.
19th century fiction is full to the gills with "What Happened to the Mouse?" scenes, partly because many books of the time were originally written for serialization in magazines. When the writer's on Chapter 24 he might forget or misremember what he wrote in Chapter 1, published two years previously. Dickens was infamous for this.
In Anthony Trollope's Barchester Towers from The Chronicles of Barsetshire, it's stated repeatedly in the first nine chapters that the new bishop will spend most of the year with his wife in London, leaving the actual running of the diocese to his assistant Mr. Slope. But the bishop and his wife never actually leave. (The reason is that Trollope put the book aside for a year, and when he returned changed his mind about what would happen without bothering to rewrite the first nine chapters.)
In the original British printing of Good Omens, it is never revealed what happened to Warlock the false Antichrist after he is taken to the fields of Megiddo by the forces of hell and revealed as a sham. For the American edition the authors added about 700 extra words revealing that he is alive and well, understandably perplexed by his experiences, and heading back to America thanks to some reality-manipulation by Adam.
Richard's two hulking bodyguards Ulic and Egan disappear from the narrative entirely after Temple of the Winds, and no reference is made to where they are, or what they're doing. Their sudden and conspicuous return to the plot in Confessor seems to suggest Goodkind actually forgot about them entirely.
There's also Jebra, the seer who first appears in Stone Of Tears. In the final trilogy, she's brought to the heroes by Shota to tell them about her experiences being caught in city conquered by the Imperial Ordernote surprisingly, she manages to avoid the usual fate of women in such situations. Shota leaves her there, but in the next book she's mentioned as having wandered off, and there's almost no effort made to find her, and she's never referenced again.
This happens with a lot of minor characters/villains/etc. throughout the series. Goodkind tends to bring in stuff strictly to serve as a plot device or MacGuffin, and then forget about it after it's served its purpose, or dismiss it with only a cursory mention.
Several WarcraftExpanded Universe novels mention princess Calia Menethil, the older sister of Big Bad prince (and now Lich King) Arthas Menethil. Calia's fate has never been revealed; in each book, she simply drops out of sight and is never mentioned again. She is the subject of several Epileptic Trees in fan circles.
Averted in Perfume: The Story of a Murderer, which always pauses the narrative to explain what happens to characters that drop out of the plot. Because the main character is a Doom Magnet, everyone he associates with dies soon after they part company.
Happens quite a bit A Series of Unfortunate Events. The Quagmires, Friday, and the rest of the island inhabitants in The End, Mr. Poe, and many, many more. Curse you, Lemony Snicket, you psycho author you. Knowing the author, this was probably completelyintentional.
In the first Sherlock Holmes story A Study in Scarlet, Watson mentions that he "keeps a bull pup" before moving in with Holmes. Once he moves in, the bull pup is never mentioned again. Maybe it died between two adventures? Though there is an explanation that's seen print is that "to keep a bull pup" is slang for "to have a short temper"—or that it's a revolver.
In Peter F. Hamilton's The Night's Dawn Trilogy, a fairly major character in Book 1, Kelven Solanki, vanishes without a trace at the end of the book after being promoted and assigned as a liaison and advisor to Admiral Meredith Saldana on his flagship. Despite Saldana and his taskforce playing major roles in Books 2 and 3, Solanki is nowhere to be seen. The author later admitted in a Q&A on his website that he had simply completely forgotten about him, but his overall importance to the story had been fulfilled. Given that the ending was so comprehensive that even the fate of a minor car thief who appeared for one paragraph is wrapped up, Solanki's abrupt disappearance seems a bit unfair to the character.
Steven Erikson's Malazan Book of the Fallen series is also noteworthy: a full list of characters who appeared briefly and then vanished would be quite long, but chief among them is Tattersail and her reincarnated form, Silverfox, who vanished along with several thousand kickass undead warriors in Book 3. Apparently their story will eventually be told by Erikson's co-writer, Ian Esslemont, several years down the road. Maybe.
One of the many things wrong with The Legend of Rah and the Muggles by Nancy Stouffer is the sheer number of mouse plots in the story. The mother of the twin protagonists, having been recently widowed at the start of the story, enters a very heavy flirtation with the palace butler before shipping her kids off to save them from impending doom; what becomes of the mom and the butler, you we never know. Later, the twins are deeply involved in the search for a specific treasure chest; when it's found, the bad twin insists on claiming it, to which the good twin consents. Not only is it never mentioned again, but the reader never even finds out what was in the chest that was so important.
Romance of the Three Kingdoms raises numerous mouse questions, as might be expected of a semi-historical narrative with Loads and Loads of Characters. To quote the book's 17th-century editor, "A beloved commander, a beloved son, lost for the sake of a woman... but what happened to lady Zou?"
In Raymond Chandler's first Philip Marlowe novel The Big Sleep, all of the various murders and crimes are explained, except that of the Sternwoods' chauffeur, Owen Taylor. During filming of the 1946 film adaptation, director Howard Hawks and screenwriters William Faulkner, Leigh Brackett and Jules Furthman sent a cable to Chandler, who later told a friend in a letter: "They sent me a wire... asking me, and dammit I didn't know either."
Reiko from James Michener's Hawaii simply disappears toward the end. She's a secondary character with an interesting plotline, but after her husband dies she's never heard from again, leaving the reader to wonder whether she ever accomplished her thwarted dreams.
Tom Robbins's Still Life with Woodpecker hangs a lampshade on this, when Leigh-Cheri's reaction to the story of the Princess and the Toad is "Whatever happened to the Golden Ball?" (that the princess was chasing when she first found the Toad.)
In Musashi, a novel based on the life of Miyamoto Musashi, the title character learns that his sister has been arrested as a ploy to lure him out of hiding. He's about to play right into the officers' hands when he's stopped by the kindly priest Takuan, who then imprisons Musashi himself for three years so he can study the classics and become a more thoughtful person. The story promptly forgets all about his sister, except for a brief mention at the end that she's moved to another region and is happily married, with no mention of how she got out of jail.
Dan Simmons's Illium/Olympos cycle. What happened to that mice colony? What happened to that humongous tentacled brain? Where did Caliban go? Did moravecs manage to get rid of those 768 black holes? Can the remaining firmaries be turned on or not? Why didn't anyone care for more than seven years? Who the hell was Quiet and did (s)he actually do anything? Has the quantum stability problem been solved? If yes, then how? Aaargh, so many questions...
The sheer amount of detail in the Harry Potter books leads to a number of these, too. Harry pulls a cracker and out come, among other things, several live mice. But mice are not throwaways like the other things in the cracker. Neither Harry nor anyone else is ever mentioned as keeping pet mice. Harry muses that Mrs. Norris got to them.
Ludo Bagman is forced to flee from goblins at the end of Goblet of Fire. He is never seen or heard of again.
Cornelius Fudge, the Minister of Magic for most of the series. His last official position is at the start of book 6, where he's been sacked as Minister, but is kept on as liaison between the Ministry and the Muggle Prime Minister. No word at all of him or how he reacted to Voldemort taking over the Ministry in Book 7.
It was never explained what was behind the veil that Sirius fell through when dying in book 5, though as it is in the Department of Mysteries, it is likely that no-one knows. Even though in the movie Sirius was dying as he fell, in the book it's obvious that the fall through the arch is what killed him. So, it's fairly clear what lies behind the arch, especially when you take into account what Harry heard from it. Luna's comments to Harry at the end of the book would seem to confirm this theory, but then again, consider the source.
It's pretty clear when you remember that "passing beyond the veil" is a common euphemism for death. Clearly the Department deals with the Big Questions: Prophecy, Death, Thought (the room full of brains in tanks, whose scroll-tentacles are specifically described as "thoughts",) et cetera.
It is never revealed what happened to Lavender Brown at the end of the seventh book. She is seen attacked by Greyback, then Hermione saves her, she was seen "feebly stirring" and never mentioned again. There is no evidence she lived or died. The movie resolved this by making her die.
Also Winky, who was last seen as an alcoholic wreck—which was probably not helped by finding out that Barty Jr. killed Mr. Crouch. She's briefly mentioned afterwards but not in the final book, though Word of God says she (somehow) got over her problems and took part in the Battle of Hogwarts.
In Thomas Harris's Black Sunday, Lander gave his pregnant ex-wife two tickets to the Super Bowl. No mention is made on if she went or what happened to her.
In Ender's Shadow, Bean is shown drawing up Ender's army. He decides to add a girl named Wu to his group. He mentions that she was a brilliant tactician, a great shooter, and did well in her studies, but as soon as her commander assigned her to be a toon leader, she filed for transfer and refused to play. No one knew why. In the rest of the book (and in Ender's Game, which takes place at the same time), not only do they never mention her again, they even make it clear that there are no girls in Ender's Army.
Early in Tom Clancy's Red Storm Rising, Iceland is conquered by the Soviet Union to allow their submarines and bombers a clean shot at convoys ferrying war material and American reinforcements to Europe, where the ground war is raging. In the course of this operation, the ship carrying the Soviet invasion force is Harpooned and strafed by American fighters, seriously wounding the captain. Much buildup is done about whether or not the captain will survive. As soon as the ship is run ashore (most of the line handlers had been killed, so it couldn't dock) the General of the invading army takes him below to the surgeon, thinking "Maybe there's still enough time." The captain is never mentioned again, leaving the reader to wonder as to his fate.
In Reaper Man, Windle is introduced to members of the Fresh Start Club, including someone called "Brother Gorper". All the other members are specifically identified as various types of undead, and most have dialogue or subsequent references, but Gorper (whatever he is) never gets mentioned again.
Pamela Dean's The Secret Country has a "What happened to the relatives" in it: The older cousins, with whom the game was usually played, had emigrated to Australia: the younger cousins were left in Illinois with other relatives while their parents were spending the summer in Australia without them, and thereby hangs the tale. At the end, after the Illinois children show up in Australia via a magic mirror, their parents decide to accompany the children back to the Hidden Land. It's a one-way trip; they know they'll never come back. The parents cook up plausible explanations for their "disappearance", pretending they're going to emigrate to Australia also and then "just lose touch". In the middle of all the preparations, no one suggests that the Illinois relatives might like to know how the kids disappeared from what was supposed to be an afternoon trip to the library, and how they got to Australia.
In Robert E. Howard's Conan the Barbarian story "Black Colossus", the princess is in command because her brother the king is being held captive and might be either ransomed or handed over to their enemies. While discussing what to do, the first point brought up is whether to enlarge the offer of ransom; only then do they discuss the attacking army, and that, partly because the captor won't take them seriously while they might be overrun. But they deal with the army, and the captive king is never even addressed again.
Many, many things are wrong with the Maradonia Saga books, but this one is particularly obvious. Several apparently important characters—including Maya and Joey's parents and brother, the grasshopper Hoppy, and their dog—show up at the beginning and then are forgotten about for the rest of the novel. Some "forgotten" characters do make brief cameos in the ending, but it's never stated what they were doing in the meantime. Was Hoppy just hanging out in Joey's pocket the whole time or what?
In the Tortall Universe, it was because of this trope that author Tamora Pierce eventually wrote a short story about what happened to the tree that became man as a result of the mage Numair turning his Evil Counterpart into a tree in the second book of the Immortals quartet.
Early on in the novel, Miles is assigned to Kyril Island as the new Weather Officer. The officer he is replacing has been there so long that he has developed a "nose" for predicting the weather, especially the deadly wah-wahs, which is far more accurate than the available equipment. Miles is briefly terrified that everyone else will notice a sudden drop in the accuracy of reporting when he takes over, but soon has a major confrontation with the commanding officer and is transferred off the island. Presumably the poor patsy who replaces him will be no better at predicting the weather than Miles, but the island is mentioned just once more in a later novel, a decade later in book time, and it's implied that nothing has changed there.
Test readers of the book were so distracted by the potential plot relevance of some money being hidden as a relatively minor plot point that the finished novel uses illicit cookies for the same plot purpose to avert this trope.
In Lucifer's Hammer, Doctor Charlie Sharps leads a group of highly intelligent (not to mention prepared and supplied) scientists out from the Jet Propulsion Laboratory, headed toward the San Joaquin Valley and shelter. Not only are they never seen or heard from again, they are only mentioned once in passing by another character, and then forgotten.
In The Angel Experiment, the main characters (who are winged humans, part bird, part human, created by some Mad Scientists) have gone years without seeing another genetic experiment like them. When they get to a secret facility in New York, they find several experiments caged up. Naturally, they set them free. You'd think they'd want to talk to them or interact with them, maybe help them find a safe place, but it never goes anywhere. Over 5 books later, said experiments don't even get a passing mention, they're never thought of again, so it's a plot thread that went nowhere and contributed nothing to the story.
School’s Out — Forever, the Flock runs into two kids in the woods in Florida. The kids claim that they were both kidnapped by scientists, both were clearly starved, and Angel claims to get strange images of water from the minds and knows that neither are ordinary children (though she doesn't think they're mutants). The kids later confess that they were held captive by Itex and were sent to find the Flock and told that if they didn't succeed, something in the woods would eat them. These kids are never mentioned again.
Saving The World and Other Extreme Sports:
The book has an entire facility full of successful experiments, including clones of Max (introduced in the book prior and herself having fallen into this trope until that point), Nudge, and Angel. It's never revealed what happened to the experiments after the facility is captured, and again the group never thinks anything of it.
Fang starts a worldwide revolution via the children that read his blog. You'd think that something like that would get a mention in the next book, but it might as well have not happened for all the aftermath there was.
In The Nightmare Factory, Dan is scarred by a creature called a Septaurus and slowly begins to transform into one (basically a family-friendly version of Face Full of Alien Wing-Wong ). Oran gives him a potion that suppresses this transformation, even though it tastes terrible. In the sequel, Rise of the Shadowmares, there is no mention of this transformation whatsoever. There isn't even any mention of the potion, despite the fact that some of the time intervals Dan has to spend between drinking it would canonically be long enough for him to start transforming again.
Peter Pan: In Wendy's personal imaginary world, she owns a wolf pup abandoned by its parents. Naturally, when she gets to Neverland the wolf appears and becomes her constant companion—or so the narration claims, since it never gets mentioned again. Surprisingly, this detail was never referenced or expanded on in any adaptations, even though the Lighter and Softer Disney version could easily have turned the wolf into a cuddly Woodland Creature and the Darker and Edgier 2003 live-action version could have thrown it into some fight scenes. (There was at least one set of illustrations (Trina Schart Hyman's) which didn't neglect the wolf and showed it hanging around at Wendy's feet in the "Home Under the Ground" scene.)
In book four Eragon and Arya wind up captured by a group of evil priests. A young novitiate appears and agrees to help them escape. He fails and winds up unconscious, while the more competent Angela comes to the rescue. Eragon insists that they take their would-be rescuer's comatose body with them as they escape the cathedral, however after this the boy is promptly dropped off in an alley and never mentioned again.
The blind Varden soldier who mysteriously turns out to be able to see magic energy disappears after being put in under vigil by Du Vrangr Gata and is never mentioned again, not even in the saga's Grand Finale.
In the Anne of Green Gables series, Anne's adoptive guardian, Marilla gets this treatment. As the Anne series continues, Marilla gets less and less mention, being mentioned briefly in several of the books. In the final book of the series, Rilla of Ingleside, it's mentioned in passing that Marilla had died many years back.
An odd subversion. While the answer is never given in either of the books - as Dodgson meant for the riddle to be a riddle without an answer - enough of his fans pestered him about the riddle that he made up an answer: "Because it can produce a few notes, though they are very flat, and it is 'nevar' put with the wrong end in front". There is also the story a copyeditor 'corrected' the word 'nevar' to 'never' and the joke was lost.
Among suggested answers, Sam Loyd's "Poe wrote on both" is probably the best-known.
Another suggestion of Dodgson's was "Because there is a B in both."
In Lauren Myracle's Rhymes With Witches, the main character's best friend's older sister is described in detail in the first few chapter. However, she is rarely mentioned after that.
Towards the middle of Reset — Never Again, the two villains, who are Asian, try to hire a detective to find the whereabouts of the heroes. It turns out, however, that the detective is a member of the Oriental Exclusion League, and says that she is going to tell their leader, one Tveitmoe, about what had happened. Neither Tveitmoe nor the detective are ever mentioned again, and the villains do not appear to be hampered by any bigots after that.
Just before the timeskip in the Thoroughbred series by Joanna Campbell, Ashley reveals she's pregnant with her second child and "due in January" (incidentally, the scene plays out almost exactly the same as did the one in which she revealed her first pregnancy). The next book (and the timeskip) comes around, the series now follows Ashley's now teenaged daughter, and...the daughter is an only child. No mention is made of Ashley's second pregnancy.
In some of Tolkien's older works such as The Silmarillion, there are several minor characters that are simply never mentioned again with no resolution, although this can be forgiven since he never completed those works in his lifetime.
W.E.B. Griffin's The Corps series has many viewpoint characters simply vanish from the narrative, especially when the series timeskips into Korea. While a couple are at least given some resolution, many simply vanish between books.
In the sixth book of the 39 Clues series, Isabel Kabra mentions that Amy and Dan's parents visited, among other places, Karachi, Pakistan, and also thought Amy and Dan visited there, although they never did. This does not go unnoticed by Amy and Dan, but it is soon forgotten and never brought up again. There was even a Sequel Series, and still nothing mentioned.
Freaky Fred runs away before the newborn army is sent to fight the Cullens. He is never mentioned again in the series, even though the novella ends with Bree mentally begging Edward to be kind to Fred if they ever meet.
This happens pretty bad in regards to that novella. For a pretty good stretch of time, there have been numerous abductions, disappearances, violent murders, suspicious fires and explosions, several residents being killed for their houses, a mall getting broken into and robbed, and an entire ferry of people getting murdered and sunk. One would think that this would get national attention under suspicion of a terrorist attack, but in Breaking Dawn, everyone has apparently forgotten about the insane amount of death and destruction that happened in Seattle. Of course, given that the story is told from the point of view of Bella, that might explain the absence of such details...
In-universe: The main characters love a Cut Short novel and want to know what happened to, amongst other things, the pet hamster.
Ironically, the same thing happened to John Green by readers of the actual novel, even after explaining in an author’s note in the novel and several times on this blog that he doesn’t know anything more about the plot or characters than that which is contained in the book. Effectively, he knows exactly no more and no less than his readers. Even when they don't believe him.
In the Blandings Castle novel Leave It to Psmith, it's eventually revealed that one of the maids is an undercover detective hired by Baxter in case there's an emergency. This is never brought up again.
Joe Bob Fenestre from The Warning, Derek from The Extreme, & Mertil and Gafinilian from The Other are all one-shot characters who are set up to be bigger players but then never appear again.
There's also the unexplained fates of recurring characters, namely Starter Villain Chapman and Loren (who is Tobias's mother for crying out loud).
One mouse in Animorphsis addressed, though - literally. The Sixth Ranger David, condemned by the team to live out the rest of his days as a rat nothlit, has a book dedicated to his return, the aptly titled The Return.
In a very early book in the series, just before losing consciousness, Prince Alloran manages to gasp out a warning that the Yeerks are infiltrating the Andalite home world. To quote Cinnamon Bunzuh!, "Well, there's an extremely crucial piece of information that we will never, ever hear mentioned or discussed again." The author admitted in a later interview that she forgot about it.
You never hear very much about Cassie and Jake's parents and the being that caused Jake to see the alternate future in book The Familiar.
Jake's parents are said to be freed after the war was over, and Rachel's mother was at her funeral. None of the parents, or Rachel's sisters, are mentioned again.
Jay shows up briefly early on and then disappears.
Kurt Schwenke somehow gets away again.
Colonel Olds is gradually built up as having a disdain for Mike that he tries to act on, but nothing seems to come of it.
Somewhere between this trope and Aborted Arc, several books of Galaxy of Fear end with a stinger where the horror of the book is still around, or There Is Another, or something of the sort. Exactly one of these is followed up on.
Eaten Alive: D'Vouran is in another part of space being stumbled upon by spacers.
City of the Dead: Zak was injected with the reanimation serum and may either be an intelligent zombie now, or will become one as soon as he's killed.
The Planet Plague: A cure for early stages of The Virus has been distributed. The lost natives of this planet recorded that cure, so they were good at coping with diseases - but it was some worse disease that wiped them out. And as he flees, the villain has some new lethal vial with him.
The Swarm: two Explosive Breeder bugs are on the heroes' ship, though that is maybe more a nuisance since as fast as they breed, they can only produce ten eggs a day and will be noticed before it gets dangerous.
Spore: Spore's tiny central pod is floating free through the Asteroid Thicket, ready to endure for centuries until it's picked up again, as Jerec's forces look for it.
Clones: Darth Vader's forces are gathering up the clones, and he thinks the ones of the Shape Shifter and those of the Arrandas - one of whom is Force-Sensitive - could be studied before they were destroyed, and the Emperor would be interested in this cloning technology.
By fifty novels in, these had built up in the Doctor Who New Adventures series. Just a few examples: What happened to the "eight-twelves" from The Highest Science? What happened to the Charrl from Birthright, last seen living in the back of the TARDIS? What happened to that TARDIS, last seen in a tarpit on a parallel Earth, following which the Doctor took the one belonging to his dead counterpart? And back in the first trilogy, didn't the Doctor leave an insanely powerful alien in the body of a human baby? The fiftieth novel, despite having a wedding to organise, manages to resolve an awful lot of them.
In The Lion, the Witch, and the Wardrobe, the animals having a dinner party before being turned to stone by the Witch (leading to Edmund's Heel–Face Turn) are never mentioned again. Aslan being omniscient and all, or perhaps Edmund told him, he probably gave them a visit to restore them. Lewis got a What Happened to the Mouse? letter from one of his readers (or the reader's mom) about the matter, and hastily wrote back a Word of God that of course the animals at the dinner party got turned back, just not on stage, and he was very sorry the child was distressed about the issue.
Then there are the servants Ivy, Margaret and Betty - though it's stated up front that they "do not come into the story much," it's more to the point to say that they don't come into the story at all. At which point it might have been better not to mention them in the first place.
Specifically the end of Autumn Twilight. It's revealed early on that hoopaks are a kender's birthright (implying they inherit them from their parents) and Tasslehoff's fighting style incorporates the hoopak heavily. He uses it both as a weapon and a distraction. Then, in the Tomb of Kith-Kanan, he says he loses it and leaves it behind in the tomb, and vows to return to get it. The sequel, Winter Night, takes place a few months later, yet for the rest of the trilogy (and the following trilogies) the hoopak is never mentioned again.
The sword of Kith-Kanan itself. Tanis comes by the sword in a very fantastic way (apparently the skeleton of Kith-Kanan "gave" it to him), yet the sword doesn't do anything extraordinary in "Autumn Twilight", and then when the fellowship is broken in "Winter Night" Laurana makes a choice to leave the sword behind with the elves so she can carry the dragon orb/dragon lance. About sixty years later, in Lost Star, Laurana raids the Qualinesti treasury and pulls out another Cool Sword, Lost Star. What happened to the Sword of Kith-Kanan?
In Les Misérables and its adaptations, whatever happened to Valjean's sister and her kids that he stole bread for in the first place? Granted, 20 years have passed and he's now on the run, but one could imagine he'd find some way to check up on them if he could. They are given some mention later on — Valjean gets some news that reveals that all but the youngest kid are gone, the sister works constantly to support them, the little boy trudges to school every day and waits for it to open, and in winter a kind lady who lives near the school lets him come in and sit next to the stove to warm up until it opens. Beyond that, they don't come up, and Hugo says he will not mention them and doesn't know what happened to the rest of the children. Some productions avoid this by having Valjean steal the bread for himself instead.
In Sing You Home, Zoe and Vanessa are fighting in court against Zoe's ex-husband Max in order to acquire the frozen embryos that Zoe and Max made in IVF treatments during their marriage. Max, who becomes a born-again Christian, plans to give the embryos to his brother and sister-in-law (who he also happens to be in love with), who are infertile. In the last chapter, Max starts contemplating just letting Zoe and Vanessa have the embryos, since he doesn't want to see his brother and wife happy together, and knows he can never be with his sister-in-law Liddy. In the epilogue, it is revealed that Max did give the embryos to Zoe and Vanessa, and is about to marry Liddy. This still leaves many questions unanswered. When did Max get together with Liddy? How did the court decide to give Zoe and Vanessa the embryos? Did Max call off the court proceedings and give the embryos as a gift? Is Max still a Christian?
Charis is sold into an indefinate term labor contract because of the fanatics seizing control of the colony on Demeter. By the end of the book, she's in contact with authorities, but no mention that they are even sending news.
Jagan's post is attacked by Jacks, but not all the people there are killed — the Company men specifically mention retrieving Sheeha. No mention of them is made in the end.
Shann is able to tell that something is wrong at the post because a man working there is not in his garden. You do not learn whether he was prisoner, killed, or escaped.
One of the very most infamous examples there is comes from Remnants. D-Caf - a major character - completely disappears after he accidentally kills Animull, never to be mentioned again.
In Astrid Lindgren's book Mio My Son, the evil of Sir Kato is so pernicious, such a blight on body, mind and spirit, that the mere mention of his name causes the sky to darken and men and animals to weep. The little prince sees a number of butterflies lose their wings. When Kato is defeated, the land, animals and people are all healed and one of Kato's servants who died to help the children even comes back to life. You can assume that the butterflies were healed too, but the author doesn't mention it.
Joe Abercrombie's The First Law universe has an excellent backstory centering on the sons of Euz and how they created the modern world. The issue is that, of Euz' four sons, Juvens (the first), Kanedias (the second), and Glustrod (the fourth) all have the important moments of their lives and deaths detailed. The third son, Bedesh, is mentioned only once, and his eventual fate is not elaborated on.
In Malpractice in Maggody, every member of the rehab clinic's staff eventually deserts the place or is called away except for Dr. Stonebridge, who's passed out drunk in his apartment, and the guard dog that's last overheard barking in its kennel. It's implied that Stonebridge will end up doing cut-rate face lifts in Mexico, but nothing's said about the dog's fate.
Early in 1632, a young woman runs into Grantville pursued by soldier who want to rape her. The uptimers kill the soldiers, but one of them is wounded in the process and once he's stabilized the girl is long gone. She doesn't appear again in the novel, but thanks to the opening of the universe to many authors, there is a short story on the subject in the first Grantville Gazette.
In Codex Alera, First Lord Gaius Sextus is married to a woman named Caria, who is actually the lover and co-conspirator of Sextus's rival, Aquitainus Attis. She barely appears in the series, and her last appearance is near the end of book five, when it's revealed that she was poisoning Sextus and had been for years. She then vanishes from the story and is never mentioned again. Word of God ended up confirming that she died when Alera Imperia blew up. In the same series, the Windwolves are major characters in the first three books, barely mentioned in the next two, and are last seen forming up for a battle against the Vord at the end of the last book. No mention is made of their ultimate fate, though some people they were about to fight alongside appear in the epilogue.
In Please Don't Tell My Parents I'm a Supervillain, the origins of the jade statue are never explained, particularly not why it's guarded by an Eldritch Abomination. Penny maintains that it's super dangerous but this is never elaborated on and she later uses pennies to transfer its curse to other people (hence, Bad Penny) without any indication of the aforementioned danger.
In Julian, a secret agent happens to see an imperial robe in a dyeshop. The increasingly insane Gallus has an innocent man executed for it, but it's never explained who the actual buyer was or why they wanted it.
It happens a number of times in Magic: The Gathering tie-in novels. Due to the novel line being written by many different authors, characters can vanish between books, but sometimes it even happens within a single novel:
In The Brothers' War, Rusko completely vanishes after the Yotians' surprise attack on the Fallaji, with not a single mention afterward.
High-ranking Phyrexian Croag is plotting Urza's death at the end of Bloodlines. He is never seen or mentioned again afterwards.
After the power of the primevals is broken in the Invasion Cycle, Treva and Dromar are never seen again. They are commonly assumed to have perished during the invasion, but this has never been confirmed.
Squee gained immortality during the events of the Invasion Cycle. He survived the invasion but has not been seen since.
Marcie, a girl in the first Origami Yoda book, is mentioned once and never again in the entire series. Given that the series takes place in a middle school and Marcie was in 8th grade, it's likely she moved on to high school and forgot about the whole incident.
At one point in Watership Down , rabbits tell the story of "The Black Rabbit of Inlé". At the end of story, god Frith waits for the hero and his trusty companion with the bag of gifts. The hero gets replacement ears, nose and tail. The reader never finds out what hero's trusty companion got.
In The Moving Finger, nothing is said about what happens to Megan's two young half-brothers after her mother is murdered and her stepfather is arrested for the crime. This is especially disturbing because she is almost definitely their closest remaining relative.
The school is mentioned to have Psi-Hounds. They are never brought up again after the first book.
Rose's Strigoi hunters in Novosibirsk never appear again after she is abducted by Strigoi-Dimitri.
In The Wild Ones: Moonlight Brigade, Basil the snake doesn't show up or is even mentioned by any of the Flealess. Which is strange, considering almost every other side character from the first novel (including Mr. Peebles, whose role was far less important compared to Basil) shows up again at least once.
In Wizard's Bane by Rick Cook an Earth programmer Trapped in Another World creates a compiler to write spells like computer programs. This allows any human to cast spells to protect themselves from magic creatures, and even should allow many to write their own spells. But locals lack the proper mindset for programming, and by the start of Wizardry Compiled there's only one case of a local genius improving Sparrow's program. Besides, that patch creates half of the novel's problems. The patch creator is never mentioned again. Somebody that good would eventually join or challenge Sparrow. Maybe he or she was in one of the villages that disappeared without a trace when immortals retaliated.
Another example appears in Wizardry Cursed. Wizards steal a powerful meteorological computer (from KGB agents smuggling it to USSR) and leave a pile of gold in exchange. The readers are left to wonder about smugglers' fate, but desertion seems a likely option. Years later they appear in Wizardry Quested as important supporting characters. They are shady businessmen who "put together aviation-related 'deals' of much import but vague content".
In Allegiant, after the peace treaty is formed, Marcus leaves Chicago and no one knows what happens to him.
In Paper Towns, nothing is said of Margo's third friend Karin, the one who had informed Margo that Jase was cheating on her with Becca, after Margo and Q leave Flowers at her house as an apology by Margo for calling her a liar.
MARZENA: The unnamed TAR Kernel, he just disappears after his Rousing Speech and is never heard of again with no concrete explanation as to where he went or what happened to him.
We never learn why Cinna requested District 12 (as he says he did in book 1) and we never find out if Portia did the same. We also have no clue why Cinna doesn't have a Capitol accent or the Capitol sense of style, despite that not making much sense if he's a fashion designer who's lived in the Capitol for his entire life.
In Catching Fire, Johanna says everyone she loves is dead. It feels like it's going to be important for her Character Development, but.... Elaboration? Explanation? Don't count on it. There's a popular guess in fanon, though: most likely Johanna's family was murdered by the Capitol, likely for refusing to be used by the Capitol after she won like Finnick was. Based on her personality and what Finnick says about his family being threatened, this seems the logical explanation.
In Mockingjay, Katniss gets a bow with "special properties." She never once mentions them again, uses them, or even explains what those properties are, besides the fact that it can vibrate to say hello. This could be the reason it's able to shoot down planes, though.
What happened to Old Cray? He somehow disappeared when Thread took over. It's not pointed out what exactly happened to him.
Why were Lavinia and her companion fleeing the Capitol to District 12? It's likely they may have been trying to get to District 13 for some reason, but how did a couple of Capitol kids come to be running away when most adults never develop the courage, or even the inclination in most cases?
Bonnie and Twill were also trying to get to District 13 in Catching Fire, and wound up being fairly close to where Lavinia was when she was captured. The last Katniss sees of them, they're successfully hiding out and planning their next move, but when Katniss and co. reach District 13 in the final book, Bonnie and Twill are nowhere to be seen. Katniss briefly Hand Waves their absence, commenting that it must be incredibly rare for those who flee to actually reach District 13... then they're never mentioned again.
Commander Lyme is introduced in Mockingjay as a former victor and leader of the rebels in District 2. She's built up as if she's going to be important somehow, but when the surviving victors have their meeting towards the end of the book, she's nowhere in sight and is never mentioned (though the reader must assume she's been killed at some point in the interim, as it is stressed that all surviving victors are present).
At the end of the Goosebumps book Night Of The Living Dummy II, Slappy is given a No-Holds-Barred Beatdown by Dennis (the protagonist's old dummy) which results in Slappy's head getting cracked open, and a large white worm-like creature crawling out of it, then escaping through a crack in the wall. Seeing as Slappy is the franchise's most iconic and recurring antagonist, it seems a bit odd that this was never explained nor brought up again. Also one of the rare literary examples of a Giant Space Flea from Nowhere.
A Story with Details by Grigoriy Oster is built around multiple aversions. In the first chapter an amusement park manager starts to tell a short fairy tale to the carousel horses. The second chapter is a story about a misbehaving boy at a zoo. Then horses start asking questions about unimportant characters, like the policeman the boy threatened or the rhinos who looked at the boy disapprovingly. The following 42 chapters are the manager's answers that create more questions and more answers. When he finally finishes and leaves in the morning, one horse recalls they forgot to ask about a she-elephant with a calf, and should ask next night.
Edgedancer (a novella of The Stormlight Archive): One character's Heel Realization is spurred by a pack of Parshmen whom the Everstorm transforms into the deadly Voidbringers, proving to him that the Desolation he's been denying has, indeed, arrived. He leaves, filled with guilt, and the story jumps into the final chapter, never mentioning the Voidbringers again. Maybe the reformed villain killed them? Maybe they ran away? But why is nobody mentioning them?