Sailor Moon started ditching the entire supporting cast of the anime (including Usagi's parents and brother) sometime after the second season. All except Usagi's mother completely disappeared in the fifth and final season. As well as her best friend Naru and classmate Umino, whose final appearances are in the direct to DVD special Ami-chan no Hatsukoi at the end of the 4th season. They actually never appear in the 5th season and were already relegated to extras by the 3rd. The manga actually ditches them even earlier.
Doctor Tofu from Ranma ½, disappeared after the first third of the manga series because his role as Mr. Exposition for weird martial arts was adequately filled by Cologne, one of the Trickster Mentors in the series. Fanfic writers keep using him to provide a second opinion or comedy relief, though there was a joke in the fan community that he had fallen into an open sewer and died. This was only in the manga; he made minor appearances in the anime throughout the series.
Rumiko Takahashi's earlier manga Urusei Yatsura does this with Princess Kurama: after some time, she disappears altogether. She continues to get small cameos in the anime because of the crowd scenes.
"Lum's Stormtroopers" disappeared from the manga after Shutaro Mendo showed up, as he more or less filled their roles of being the rival to Ataru. The four became more significant in the anime.
Happens egregiously in the Slayers novels, more so the Special series. The most notable example is a mercenary named Lantz, who helps with the fight against Copy Rezo in lieu of Amelia in the third novel. He literally runs off, never to be heard of again. The only novel-exclusive character who is returned and remembered is the swordswoman Lemmy, who manages to make it to Amelia's side story and a radio drama. In the anime, similarly, Amelia's uncle, Christopher, doesn't get a single mention again after Slayers Next.
The Steel Saints from Saint Seiya, characters original to the anime's first season, completely disappeared before the 12 Zodiac Temples Arc. A common joke among the fandom was that they took the wrong plane from Japan to the Sanctuary.
Shamal from Katekyo Hitman Reborn!. He got a passing mention in the beginning of Future Arc and is never seen or mentioned anymore.
Several major characters in Medabots, such as Dr. Aki's niece Karin, never showed up in the second season. That said season was made specifically for the US market may have something to do with it.
In Onani Master Kurosawa, the unnamed younger sister of the protagonist appears in some boxes trying to talk to Kurosawa, only to be solemnly ignored (even by the author, after some chapters).
A character named Hanasaki appears. The characters become friends with him, he's around for some chapters, but after the Death-T arc, he disappears and is never mentioned again.
Miho Nosaka. She was a very minor character and Honda's love interest in the manga. She gets promoted as a main character in the first anime series, but is not in the second series Duel Monsters. But she was referenced in GX when she was listed among the missing Domino residents who are sent to the World of Darkness.
For a Hentai with so few characters, it's quite noticeable when Io Azuma in Moonlight Lady simply vanishes.
This happens a lot in car racing manga. In Wangan Midnight, literally dozens of minor characters, including Rumi Aikawa, Ma, Kochan, Yoshiaki Ishida, Harada, and Makoto Morishita, have been dropped without so much as a footnote.
Kimi Toudou, Kawaiiko and shameless Gold Digger from Fruits Basket. The last scene we see with her is her interrupting a love confession between Yuki Sohma and Machi Kuragi while looking for her missing hair brush. She never appears again. This is somewhat surprising as virtually every other minor character in the series makes some sort of appearance for a Cleaning Up Romantic Loose Ends chapter at the end. (Even Naohito Sakuragi, equal to Kimi in terms of importance, gets paired off with Motoko Minagawa [the Yuki Sohma Fan Club President] at the end.)
Klaus in Hayate the Combat Butler disappeared for around two hundred chapters with absolutely no explanation. Once or twice, it was helpfully noted that, yes, he's still alive.
In Mahou Sensei Negima!, Jack recounts the story of how Nagi saved his love interest, Queen Arika, a few years before Negi was born. No one has mentioned her since, and Negi hasn't asked. This also leads to a touch of Karma Houdini, since Arika has (evidently) never intervened to save her sister, her lover, or her 10-year-old son from fates worse than death.
The main character, Tamotsu, from the first Boku No Pico OVA is never seen or mentioned in the second or third ones.
Lin's puppy, Pel, from Fist of the North Star disappears some time during the Souther arc with no explanationnote only in the anime; it never even shows up in the manga. It shows up one last time in a later episode, only to never be mentioned again.
In From Eroica With Love, the first volume starts out with the main characters established as three psychic powered teens Sugar Plum, Leopard Solid and Caesar Gabriel. By the end of the first volume, they're no where to be seen and never heard from again, in favor of the actual interesting characters, Dorian Red Gloria and Klaus Heinz Von Dem Eberbach.
Because the movies are made before the actual series are released, certain characters are left out of the Pretty Cure All Stars movies. Among them, Eas/Setsuna in DX 1 and Seiren/Ellen and Ako in DX 3note It should be noted, though, that Setsuna and Ellen were evil at the time, while Ako hasn't been officially confirmed as Cure Muse, and was still busy keeping up with the silent protector disguise. Oddly enough, Itsuki and Yurido show up in DX 2 as brief cameos, despite Yuri being introduced three episodes after the week the movie was released. note Although Yuri is noticeably more OOC, not acting like a cold fish she started out as in series proper.
While most of the characters from Beyblade's first season came back in season 3, a few players didn't. Steve from PPB All Starz is replaced by Rick who took Michael's place, and Ivan/Ian isn't a member of the new founded Blitzkrieg Boys (but he's shown in a flashback with the Demolition Boys). Team WHO (the Dark Bladers) were never mentioned, and the Majestics were kicked out by the Barthez Battaillion, and only Ralf/Robert and Johnny got the luck to appear at least in one episode. The teams from season 2 also never appeared in season 3.
Lunch from Dragon Ball Z, who disappears along with Tien's first death during the Saiyans' appearances. The author admitted he had forgotten about her.