A Series of Unfortunate Events: The wart-faced man from Count Olaf's troupe disappears after the 1st book and is never mentioned again. He also didn't reappear in the movie that was made 5 years later.
Animorphs: Hey, remember Mertil, the Andalite that had also survived Elfangor's ship's crash and had been living on Earth for the entire war? Remember how he was found by the kids and decided to remain in hiding? Or maybe you don't, because he was only in one book. Having another Andalite on board would've been very useful for the kids. Even if Mertil didn't join the kids for his own reasons, it would've been nice for him to get another mention, seeing as an Andalite war hero living on Earth is kind of a big deal.
In "Cheaper By the Dozen", Mary, the second eldest child dies at age 6. This isn't mentioned in the book. She simply stops appearing.
In The Baby-Sitters Club, this was the eventual fate of most of the girls' non-club friends, fuelling speculation that the girls were disturbingly cult-like...The major exceptions are Laine Cummings, Stacey's on-again, off-again best friend from New York, and Sunny Winslow, Dawn's Troubled, but Cute best friend from California. But Sunny started her own baby-sitting club, anyway, so she doesn't really count.
Don Quixote: In Chapter I, Part I, Cervantes mentions the people who lived in Don Quixote’s house: his niece, his housekeeper and a lad who helps them with the field and the marketplace... we'll never see or hear anything about that lad again. Obviously, Cervantes had completely forgotten about this character, and didn't want to write him even in the Second Part of the novel, but in his defense, one of Don Quixote's themes is about how silly it is to detect errors of continuity in a silly fictional tale...
Near the end of Frankenstein Victor notes that, with his wife, baby brother and best friend murdered and his father dead from grief, he has lost every important person left in his life. ...Except wait, didn't he have another brother named Ernest? Did the monster get him too?
Rifleman, Stile's ally and mentor in the ways of Citizenship in the first Apprentice Adept trilogy. When the second trilogy starts up, he's nowhere to be found or mentioned.
Tobias Gregson, the only member of the main cast from the first Sherlock Holmes story, A Study in Scarlet, to never become a regular. There are mentions of "Gregsons" in later stories, but they are very few, only sharp eyes will catch them, and there is no indication that any of them refer to Tobias.
In the Artemis Fowl series, Doodah Day. At the end of Lost Colony, he is mentioned as having started working as a private detective along with Mulch Diggums. The Time Paradox takes place mostly in the past, so it's understandable that he wouldn't appear, but Mulch is actively involved in The Atlantis Complex and not only does he not seem to be working as a detective, there is absolutely no mention of what Doodah is doing.
Minerva Paradizo. At the end of book 5, she is stated to have spent 3 years obsessing over Artemis, waiting for him to return, and she was set up as a very obvious Love Interest. Two books later, and she hasn't been mentioned once since then.
The 19th-century penny dreadful Varney the Vampire began with three children in the Bannerworth family — Henry, Flora, and George — but George is never mentioned again after Chapter 36.
The Sisterhood Series by Fern Michaels: This has happened to some characters. Private Detective (former FBI agent) Mark Lane seems to pretty much vanish off the face of the Earth by the book Final Justice. It might be due to the fact that he acts as a source of information for Jack Emery, who by then is getting information before Lane does, and Jack doesn't really need him anymore!
There were two sequels to Harriet the Spy. Sport and Janie, Harriet's best friends, get not a single mention in either of them.
As the Moomimtroll series progresses, the Moomins keep acquiring new house-guests, until the second-to-last novel Moomimpappa at Sea, where, with the exception of Little My, they all vanish without a word. Particularly jarring is the Snork Maiden, who up until this point has been Mooomintroll's G-rated love interest.
Happens to the wizard Radagast the Brown in The Lord of the Rings. Messengers are sent to his house mid-way through the first book, but find it deserted. His absence is never explained, and he is never mentioned again. When asked about this in a letter, Tolkien said that even he wasn't sure of Radagast's fate.
In Remnants, D-Caf suffers a What Have I Done moment after accidentally killing Anamull in the fourth-to-last book and swears that he'll never release his worms again. Well, it would have been hard for him to—he's never mentioned in the rest of the series, even in the Distant Finale when just about every surviving character got a throwaway reference.
Happens a lot in Robert Jordan's The Wheel of Time, even to major characters. Characters like Lan and Thom disappear for whole books at a time, or come back for no real purpose and don't get much page-time. Of course, with a series that has so many characters, this is not entirely unexpected.