A Certain Magical Index was making plot threads in the last episode of the season. And the spinoff is apparently based in the same time period as the original, not a sequel that happens to have a new protagonist, although a sequel has since been made.
A Certain Scientific Railgun overlaps with Index's plot somewhat, but it's never outright mentioned in the series that it actually starts before the events of Index. The only thing that hints towards this is Touma's appearance later on, where he talks to Mikoto in a way that one would talk to someone they don't know. This implies that he's since suffered the after effects of stopping Index's Dragon's Breath and activating the Feathers of Light, which resulted in him losing all his memories.
An event with Mikoto and one of her clones is supposed to occur, which then involves her finding out about the rest of them, which immediately follows into the appropriate episodes of A Certain Magical Index. Railgun has not reached this point, so it's quite a while behind Index.
Anatolia Story has an early arc that dealt with the imminent war with the Mitanni kingdom and Queen Nakia sending her servant to acquire the 'Dragon Eye' during the tussle. The 'Dragon Eye' is found, lands in the street and Urhi is about to grab it, when Prince Mattiwaza distracts him. Urhi loses sight of the 'Dragon Eye' and it's never mentioned again.
Aoi Hana tends to have this going on. Remember Haruka's confusion about her older sister's relationship with another woman, who also happens to be a teacher at her high school? Well, Haruka herself surely doesn't seem to, and she is now mostly seen goofing off in the drama club room. The same thing seems to be going on with Kyouko's feelings for Yasuko, after Yasuko got Put on a Bus.
Baccano! manages to play with this. At one moment, one of the immortals is seen perfecting the Grand Panacea by crushing a mouse. When it is resurrected, he knows he has perfected the Cure-All Elixir. Soon, a fire consumes his workshop, allowing the mouse to escape. The final episode implies that even the mouse was a main character and continues to live to this day.
In Bakuman。, after the main characters' first manga gets canceled, their assistants are out of work. One of them gets serialized around this time, and another finds work as his assistant. The third one has a wife and children, which is explicitly pointed out as a reason why this is quite bad for him, but we never find out what he does.
Bleach: Ashido might be this manga's only genuine example of this trope. Kubo created the character and fully intended to use him but ran out of time to insert him into the manga. The anime decided to create a filler arc based around him and insert him where Kubo had originally intended to use him. However, the filler arc ended with him facing down the enemy to give the group time to escape and Rukia vowing to go back to Hueco Mundo to search for him. She never did, so Ashido's fate remains unknown.
In Chobits first episode when Hideki finds Chii and carries her to his apartment, he doesn't notice a data disk that falls from her; the close-up given to disk on the floor implies that it is something important but its never mentioned again nor it contents revealed. Another example where it's resolved in another medium; in this case, the manga has the explanation, overlapping Aborted Arc. At that stage in the show, they were essentially copying the manga into anime form, but since the anime ended differently than the manga, they decided not to use the original purpose of the disc.
Sunohara from CLANNAD, being the protagonist's best friend wasn't enough for him to play a major role or even appear more than just a few seconds in the later years of After Story, too bad for Tomoya his friend wasn't visible enough to help him face all those family problems; his only known feature during these harsh times, it's him working at some kind of a driving company.
Sunohara deserves a special mention because his status in the original Visual Novel; while the anime has all the girls together and some of their routes glued in despite the obvious Nagisa end, the Visual Novel is straightforward in making any girl not needed for a particular route disappear just like that, but Sunohara is present in all of them no exceptions thus is quite jarring seeing Tomoya suddenly not having his ever present friend when he needed him the most.
In the Kyou Chapter, all girls except for Kyou and Ryou seemed to have totally disappeared. Sure, the focus is Tomoya's blooming romance with Kyou, but does that mean his friendship with the others would just disappear like that? Same for the Tomoyo Chapter.
This is because both take place in alternate worlds based on their routes in the Visual Novel. His friendship didn't disappear, but instead never existed in the first place.
In general, Death Note manages to resolve most of its plot lines, and most events that occur early on have some sort of bearing on future events. That is except for Misora Naomi's arc in Volume II. Her death and disappearance as the result of accidentally meeting Light, who tricks her into revealing her name serves as only a minor detail in L's investigation (it simply furthers his interest in Raye Penber's list of suspects) and is never mentioned again throughout the entire series. Of course, we know what happened, but the investigation team seems to lose interest pretty quickly (which is probably natural, given how well it's been covered up). Still, it's one plot point that, all told, has little relevance to the whole.
That, and at the end of the series, we never learn what happened to Misa following Light's defeat, but Word of God confirmed that she committed suicide after finding out about his death. The anime hints at this by having a shot of her looking very blank-faced stood on top of a building. We don't see her jump... but the imagery alone is pretty depressing.
It wouldn't be quite as egregious if the "Dark Ocean" introduced in "HMV" (Dagomon's Call or Call of Dagomon in Japanese) wasn't mentioned or revisited in... let's see... six more episodes, at least. But D(r)agomon is only mentioned once more, in episode-30-something. The questions still aren't resolved, though.
Same series you also never do find out what Daemon and associates actually want with the dark spores.
In the manga version of 02, Daemon is actually working WITH Oikawa, rather than against, and his purpose for wanting the seeds is explained.
Averted quite literally during the Red Ribbon Army Arc of Dragon Ball. While immobilized by General Blue's paralysis technique in an undersea cave, Goku is saved when a mouse suddenly runs out and frightens his opponent into canceling the technique. Later, after recovering the Dragonball and with the cave crumbling around him, Goku goes back for the mouse out of gratitude and manages to escape with it at the last second. The mouse was last seen drying itself off on the beach of Master Roshi's island where presumably it lived the rest of its life in peace.
Near the end of the "Lullaby" arc, Natsu has a fight with "Grim Reaper" Erigor, the head of the Dark Guild Eisenwald. Natsu wins, but during a moment of distraction, Erigor goes missing. The characters even ask where he is, until more pressing concerns distract them again. We still have no idea where the heck Erigor is, and there doesn't seem to be any indication that Erigor will be mentioned again. Of course, Mashima-sensei has a talent for remembering little details and characters he's used in the past, and bringing them back later... but usually there's some evidence that he hasn't forgotten, even if it's just an Easter Egg. Erigor has literally never been mentioned again.
He makes a brief reappearance in the anime during the early part of the Nirvana arc. For a rematch with Natsu. He gets CRUSHED in the span of about 3 minutes.
In the Laxus arc, Natsu (and Gajeel) can't pass through Freed's barrier, which is set to block (a) people over 80 years old and (b) stone statues. This is built up as a major mystery — but Levy ends up dispelling the barrier without finding out what the problem was, and infuriatingly, it's never been brought up again.
However, it is possible that the reason for this is because Igneel, who is a lot older than eighty, has been inside Natsu the entire time, so it's possible that the same is also true for Gajeel (and perhaps Wendy). Not that that's been explained either, but...
In Family Compo, Akane Fujisaki. She gets her own Story Arc, spanning several chapters. She appears in the first pages of the new arc and calls the main character a bastard (long story). And then she drops off the face of the Earth, never to be seen again.
Full Metal Panic! has Sarah Miller (Known as Kudan Mira in the original japanese version.) People who have watched the entire series may be wondering, who the hell is Sarah Miller? Well, she's the red haired girl who is rescued by Sousuke in the very first episode. Her kidnapping by Gauron (along with her unique hair color) implies that she is a whispered and therefore fairly important, but she is never seen again or even mentioned outside of that opening scene. In the anime at least. In the novels she comes up again, where it is revealed that she is indeed a whispered (and under the protection of Mithril.) Certain things, such as the appearance of Tessa's brother imply that the anime had no idea how many seasons it would have, but that doesn't make Sarah's complete disappearance any less odd.
The Zakenna butlers live throughout both seasons of Futari wa Pretty Cure and are never shown being killed, yet their fate at the end of Max Heart is left unaddressed. They are last seen at the Dusk Zone's mansion.
Game X Rush has this whole set-up about Yuuki's past, particularly concerning his past as an assassin and how he acquired his assorted skills... which are later all but forgotten about completely, and no mention made of them again.
In-universe in Monthly Girls' Nozaki-kun, Seo starts recalling a time where she played with a boy called Okada, and how one day she put him in a back of a truck and said truck suddenly started moving... Seo offhandly ends the story there, realizing she's telling the wrong one, and she never reveals poor Okada's fate.note The answer was only revealed in the official fanbook, where it reassures readers that the truck went to a strawberry farm and Okada safely returned, and even gave Seo strawberries as a souvenir.
Ghost Hound: Remember Chiyo? Miyako's friend who sold her out to the cult? The one who was repentant and left crying at Makoto's house in the end? The house that got destroyed in a landslide?.
Initial D usually does a fairly good job of at least kinda giving closure to most of its minor characters (or reviving them for future arcs, a la Kai Kogashiwa), but there are two notable instances where Shuichi Shigeno was just plain brutal. First was Sayuki. After Takumi Fujiwara's duel with Impact Blue, they're seen at a water park, and it's strongly hinted that she has feelings for him. We'll never know for sure, unfortunately, because this is the last we ever see or hear of her. Second was the Night Kids. It's strongly hinted that they were a well-respected team before the Red Suns and Takumi came along to make a mess of things, and there are very palpable rumblings of dissention from Shingo Shoji. We see Takeshi Nakazato, once a dominant racer, lose to Seiji Iwaki, marking his third straight loss and second straight on his home turf. Seiji proceeds to publicly humiliate the team. And THAT'S IT; we never hear anything about Takeshi, Shingo, or the Night Kids ever again. (Oh yeah, Seiji also disappears into the mists a few chapters later.)
A literal example in Kill la Kill: Dr. Matoi has a pet mouse on his shoulder that simply stops appearing after a while.
An example that was caused by editing away a not-so-superfluous scene: Macross: Clash of the Bionoids was an American VHS release of Macross: Do You Remember Love? with an English dub and over 20 minutes of running time cut. One of the cut scenes was that of Hikaru contacting the bridge of the Macross after the explosion of Boddole Zer's ship, leading to a case of "What Happened to the Main Character?"!
The Macross series itself has a big one in the form of the Megaroad-01 Fleet, which was carrying the protagonists from the original Super Dimension Fortress Macross. According to the series chronology, the fleet simply vanished four years after its launch, somewhere near the center of the galaxy. Nobody, in-universe or out, knows what happened to them. The manual for the video game Macross M3 claims that the fleet got swallowed by a black hole, but fans are divided as to whether or not to take this as canon.
One of the earliest examples appeared to be that Chao claimed to be a martian from the future, and attempted to break The Masquerade in order to prevent a future tragedy. After she's defeated, she returns to her own time, and the whole "future tragedy" thing wasn't mentioned again. Until it turned out to be a massive Brick Joke with The Reveal that Mundus Magicus is on Mars, and the implication that its destruction is the tragedy she was trying to prevent.
There was also another one in the form of Zazie's appearance in the middle of this splash page in chapter 1, in which she has claws. This is never commented on or mentioned in any way in the story itself, leading to all sorts of fan theories and the eventual assumption that the author forgot about it. Untilchapter 298: She's a demon girl, although the reveal isn't all that surprising by the time it happens as the magical world is full of hybrid girls, from catfolk to dragonfolk.
The whereabouts and activities of Graf Herman remain unclear, even though he survived his battle with Negi and promised that they would meet again. He suggested that he may know how to reverse the petrification cast on the inhabitants of Negi's village when he attacked it some time before the main story begins, but by the "Where Are They Now?" Epilogue, Konoka went ahead and cured them anyway and he vanishes completely
Negi's mother, Queen Arika, despite being incredible important and surviving the events that destroyed her country, is never mentioned again. Her reasons for abandoning Negi, or her inability to help, for the less cynical, are never revealed. Similarly, the significance of Negi being her heir and what he does about it is never touched upon.
During the "Where Are They Now?" Epilogue, no mention is ever made of Anya or her fate. Interesting, as almost every other character is addressed, albeit sometimes indirectly.
Fate's Ministra Magi and the Master of the Graves also vanish right before Negi's final confrontation with Fate, and they are never shown or mentioned again. It's specially baffling in the case of Shiori, who had also become one of Negi's own Ministra.
Mobile Suit Gundam 00: During the first season, the politicians of the major power blocs played an important if supporting roles. But by season 2, there's no mention of them after the time skip. The Union President does make a cameo at the season finale.
The same thing occurs to Celestial Being's Observers, who apparently monitor the organization in one episode. They're never seen or mentioned again in the show.
Wang Liu Mei talks about finding the countries that sponsored La Eden terrorist group, but they are never referenced ever again.
In Mobile Suit Zeta Gundam minor antagonist Gates Capa completely disappears after his "sister" Rosamia is killed without being given an on-screen death himself. Though this is shown to have some kind of mental effect on him, he never appears again in the series. It's a common belief that he was meant to appear in the sequel Mobile Suit Gundam ZZ at some point. In the Zeta novels, the psychic backlash he receives when Rosamia dies kills him outright.
Played with in Monster, when we're left wondering what happened to Tenma's date for a good third of the show. Eva had a talk with her.
In The manga, there is a fight where Orochimaru summons two giant snakes. One is trapped in a swamp jutsu and later killed. The other injures Naruto and then is never seen again (not even as a puff of smoke which would indicate that it was de-summoned). The anime "corrects" this by having both snakes entrapped in the swamp; after attacking Naruto, the snake inexplicably sinks.
Summons in Naruto automatically return to where they came from eventually, so one can assume that's what happened to the second snake in the manga.
Another comes a little while later: Hayate Gekko, the proctor for the third part of the chuunin exams, is killed by Baki when he finds out about the Sand and Sound village's plan to attack Konoha. When his body is a found, a female ANBU member named Yugao Uzuki turns out to be Hayate's lover, and swears revenge. She's never seen again, and when Baki and the whole Sand village pulled a Heel–Face Turn you're left wondering how she responded. One of the video games finally resolves the question, as if you start a match between Yugao and Baki, she will declare her desire for revenge at the beginning, and when using her special move.
Filler in the Shinobi World War arc goes into this a little she's now a medic nin and gets called out to deal with Zombie!Hayate.
The Akatsuki rings. Deidara, Orochimaru and Zetsu were all used to play up how incredibly important they were and the camera lingered tellingly on it a few times when Itachi first came to town. Not a single mention otherwise, and the series ended with that particular plot point never being visited again.
There's also Hanabi, Hinata's sister, who is supposedly stronger/more promising than Hinata (which is before Hinata Took a Level in Badass, so it's unclear where she stands now), and was possibly favored for inheriting the clan. Since the arc of her first appearance, she's yet to do so again to further the Hyuuga relations/hierarchy subplot. For Pain's Invasion, she is implied to be away with her father for an unknown reason, but while Hiashi reappears to fight in the Fourth Shinobi World War, Hanabi has not.
Her absences are addressed in two filler episodes focusing her on point of view that aired right before the Grand Finale movie, The Last: Naruto the Movie, where she has a more central role. For the Pain's Invasion she was outside the village for a diplomatic outing with her father and for the Fourth Shinobi World War she is shown seeing her father and Hinata off while she protects rest of the clan in the village.
A minor example is Sasuke's Leaf headband. In their battle at the valley Naruto made an incredible fuss so Sasuke would wear it. After Sasuke leaves, one would think that an item, that represents his best friend's status as a Leaf shinobi, would be kind of important to Naruto and mentioned at least once during the long sessions of thinking about Sasuke in bed, but the headband was never shown again aside from two filler episodes.
Naruto finally gives it back to Sasuke in the second to last chapter, 699
What ever happened to Anko Mitarashi? Sasuke used her Curse Seal to revive Orochimaru a while back in the manga, several YEARS in real-world time... and just left her on the cave floor. The wiki stated that she was unconscious, not dead, and she has never appeared again, nor has any mention been made of her fate.
And Yamato? Also kidnapped because of his abilities and then... nothing. The current theory is that he's inside the Spiral Zetsu and being used to power his Mokuton jutsu.
Yamato was held inside the Spiral Zetsu before being freed and then caught in the Infinite Tsukuyomi, before being freed from that with everyone else. Anko makes a very, very small cameo in the final chapter, 700, and Yamato makes a brief cameo at Naruto and Hinata's wedding in The Last: Naruto the Movie.
What exactly was that technique? At the beginning of the Gaara rescue are Jiraya pulls Naruto aside and tells him to not use that technique. Naruto looks downcast and replies with "I know." And it is never mentioned again. Hundreds of chapters later, and nothing has come of it yet.
It is eventually clarified this was specifically using the Kyuubi's chakra due to his lack of control.
The ending of the manga leaves the fates of some of the villains in the air. Orochimaru and Kabuto are both seen being freed from the Infinite Tsukuyomi along with everyone else and are simply never seen or mentioned again. Orochimaru returns in Manga/Naruto Gaiden, however, still up to his old tricks.
The finale never really goes into detail explaining what happened to the tailed beasts after the war ended. The Chibaku Tensei prisons are broken, but nothing more is shown. From The Last, we know that Kurama happily and willingly returned to being sealed inside Naruto and the Epilogue reveals that Bee reunited with Gyuki. Also,  reveals that Shukaku and Gaara remain together, but that's it.
In One Piece, Gin has never been seen again after the Baratie arc and it is never explicitly stated whether he died from Krieg's poison. What makes this particularly odd is that Oda, the author of the series, occasionally uses cover pages to give the readers updates on characters who have not been seen in a while, but Don Krieg, Gin, and company have never been featured on any of these pages.
Talking about cover pages, Coby had his own cover page story which ended with Morgan, one of the series' earliest villains, escaping from the Marines. His next appearance was on the following, but completely unrelated Jango cover story, where he is seen taking a nap on a boat in the middle of the ocean. He hasn't been seen ever since and his fate is still unknown.
Actually Coby was given a major makeover and promoted to Monkey D. Garp's assistant. Coby has even take a level in badass in gaining serious Haki of his own, with Garp implying he will mentor him, as of the end of the Battle of Marineford Arc.
The Absol Hunter J captured in her introductory episode in Pokémon for some reason was actually not reverted back from its statue form at the end of said episode unlike Pikachu, Meowth, and Gardevoir.
The well-dressed man who trades Ash for his Butterfree on the S.S. Anne seemed to be left for dead. The last we see of him he's standing by the trade machine after trading Butterfree back to Ash while Ash runs after Butterfree's ball, which slipped from his hands. Not long after that the ship capsizes. Considering we don't really know where the trade machine was on the ship, he may or may not have had time to escape. It's possible there's a well-dressed corpse somewhere in that ship.
In "Go West Young Meowth," young Meowth runs across the street and causes a huge traffic accident. We do not learn the fate of the drivers, but it's likely at least one of them died.
In the very first episode, Ash borrows Misty's bicycle and gets it trashed. For the next few episodes she justifies her trailing after him by demanding he replace her bike, but by the end of the first season everyone's forgotten about it, even Misty.
Misty gets her bike back from the Viridian City Nurse Joy, whose predecessor repaired it for her, in the episode "Gotta Catch Ya Later".
The GS ball. Delivering this Poké Ball and finding out what is inside was a major plot point for two seasons and was the initial trigger for the entire Orange Isles arc. Once it reaches Kurt (who can't open it, but promises to keep trying) it is never mentioned again. Unfortunately, this is the result of an Aborted Arc - the ball was supposed to contain the time-traveling Pokémon, Celebi, but when it was decided instead to make Celebi the main focus of the fourth movie, the GS Ball was dropped in the hopes that everyone would forget about it. We didn't.
Whatever happened to Jigglypuff?
She(=)ens up finding a Whismur in Season 6, whose ability Soundproof allows him to listen to the entire song of Jigglypuff. Whichs ends Jigglypuff's quest for finding someone who could listento her song without falling asleep.
In one of the two released pilots of Sonic X, there is a very short clip of a glowing silver hedgehog looking at the camera quite smugly, implying he is some kind of villain. Absolutely no mention or reappearance of said character occurs throughout the entire running of the series. A picture of the hedgehog appeared on the official Japanese website of the anime, with the filename Nazo.jpg, with the word "Nazo" meaning "Mystery" in Japanese.
The battle that was being set up at the end also never evolves beyond a MacGuffin. For that matter, the war itself.
Automatic health regeneration in Sword Art Online. Kirito uses it to demonstrate that other players can't hurt you if you're many levels above them, because your health regenerates so quickly that they can't damage you fast enough to kill you. At the time, it appears to be a standard ability of every character. But after that episode, no one's health regenerates on its own. This is especially noticeable when Kirito drops to very low health in a boss fight and when a psychotic mole in his new guild nearly stabs him to death; in neither case does his health regenerate at all.
Tales from Earthsea has plenty, being an adaptation of several books at once. At the very beginning of the movie, Arren stabs his own father and takes his sword. While the sword becomes a plot point, the exact reason why he stabbed his father is not explained apart from some sort of "desperate rage". While Arren's fear of death (or rather, his life's insignificance due to everyone dying some day) leads to him becoming kind of crazy and suicidal at times, it does not explain the murder. Also in the opening scene, sorcerers are shown to lose their power and dragons fight in the sky. The king, before being killed, is informed of some sort of plague spreading and ill omens of "dragons returning from the West". Sparrowhawk is seen losing his powers as well, and says it's due to "the Balance being upset". The Balance is briefly mentioned to be between earth and sea, fire and wind (the dragon's established element), life and death. None if this is mentioned again. The plague is never explained, sorcerers are not shown to regain their powers even after the Big Bad is killed, and the dragons don't appear apart from one main character briefly turning into one at the climax.
In Tamako Market, there is the subplot of Midori possibly having deeper feelings for her best friend Tamako. In one episode, Midori even tries to prevent Mochizou from becoming closer to Tamako because of this. None of this is ever mentioned again later.
The very first scene of Tenchi in Tokyo features a pair of mysterious characters who appear in a flash of light, break into a shrine and steal a mysterious and seemingly very important box. Who are they? What's inside the box? Why are they stealing it? Your guess is as good as mine, because neither they nor the box are ever seen or mentioned again.
Speaking of Tenchi, in Tenchi Muyo!, when Sasami is introduced, she's seen with a ferret-like creature by her side. A few official works even show her with it. However, the creature's never seen again after Ryoko crashlands everyone back into the Masaki Shrine and is replaced by Ryo-Ohki. The compilation manga Densetsu no Sasami (Sasami Stories in America) have a set of comedic yonkoma that show the creature reappearing, gaining the name "Okiyama" and fighting a one-sided rivalry with Ryo-Ohki.
More than that, what the hell happened to Dilandau / Selena? Was he/she cured? Is she/he still insane? Is s/he the Emperor / Empress?
Dryden's mermaid is returned to the ocean, presumably to her people. It's explained in a short line about how he's sorry for keeping her locked up in a tank for so long. Still, blink and you'll miss it.
In the earliest chapters of Wandering Son several children hang around Takatsuki, Saori, and Nitori. By volume 2 only Kanako is left.
Maho originally befriended a girl when she became a model. As Anna and Maho went from enemies to best friends the girl appeared less and less until she was dropped.
At the end of Wolf's Rain, all of the wolves are seemingly reincarnated as humans except for Blue, who is never seen or mentioned again.
In Yu-Gi-Oh! 5D's, third season character Sly, a friend of Ruka's from school, tried stealing Stardust Dragon and made that his personal goal by the episode's end. Nothing ever comes of this.
In an early arc, Yusuke and Kuwabara enter the running for the honor of being Genkai's apprentice, culminating in a tournament. During the tournament, two mystery contestants (the young Shaolin and Chimpo) face off in a completely dark room. The viewers see nothing of the fight until the Shaolin comes back, and Chimpo... doesn't. He's never seen or referred to again, so no one really knows happened to him. His absence is so ignored by the other characters that "Nobody remembers Chimpo" becomes a Running Gag in Yu Yu Hakusho Abridged. Although, Genkai explicitly states that anything goes and they can kill their opponents, so....
After Hiei defeats Bui, he leaves him lying in the stands, refusing his request to finish him off. Bui is never seen or mentioned again, although it's possible he died when Sakyo self-destructed the stadium. Similarly, Mitamura's three students are never seen again after struggling against Toguro's sucking souls to maintain his 100% form, without even any suggestion that they succumbed or survived.
Also, in the anime, the "nurse" of Team Masho, Ruka, who traps Hiei and Genkai in a barrier for the duration of that round, never shows up again after the barrier wears out. In the manga, she shows up in the epilogue going into an Earth modelling business with Koto and Juri.
A rather literal semi-example of the trope: in Zipang a minor background character apparently considers a mouse living in temporary barracks as his pet, and is concerned when it doesn't show itself, which is a prelude to an artillery strike about to commence on the base. Several episodes later he is seen walking among survivors, holding a mouse, but no comment is made about him or the mouse.