Sequel Non Entity

Sometimes a character from a movie doesn't appear, has a brief Hand Wave explaining their absence or even does not get in any way referenced in the sequel because the actor didn't want to return, or couldn't for scheduling reasons, and the producers feel (maybe justifiably, maybe not) that nobody would care about the lack of resolution, even if the character ended up connected to the major characters in an important way. They may have appeared as a minor joke character early in the previous film, whom a main character offends somehow, but returned at the end turning out to be their new Mother-In-Law. Logically they'd now be important to the character's future, but when the sequel rolls around there's either a very brief Hand Wave over their non-appearance, or simply no reference to them at all. Or the character may have been a love interest or living MacGuffin who the producers felt wasn't popular or memorable enough to recast, deciding the only people who would care would be nerds.

See also Brother Chuck, Demoted to Extra, Put on a Bus, Estranged Soap Family and Sudden Sequel Death Syndrome.

Film

 * Rebecca from Night at the Museum doesn't appear or get mentioned in the sequel, despite starting a relationship with Larry at the end of the first film, because Carla Gugino was busy filming Race to Witch Mountain. Not to mention it'd be weird to have her along with Larry's relationship with Amelia Earhart.
 * In Men in Black II they mention Dr. Laurel Weaver (Linda Fiorentino) once just to "explain" why she isn't with them and then drop the whole thing.
 * They briefly mention in the second XXX movie that Xander Cage was (allegedly) killed and that's why there's a new xXx. There was even a scene filmed with Cage being blown up in an attack from the sequels villains, resulting in the tattooed chuck on his neck landing in front of the camera, just to thoroughly avert He's Just Hiding. Smart choice not to include that in the final cut, since they're planning to bring Xander back for another movie.
 * They briefly explain that the Keanu character from Speed broke up with her because of how their relationship started.
 * Nightcrawler's absence in X-Men: The Last Stand was Handwaved in the X2 video game, so the die-hard X-Geeks knew. All us normal people were left in the dark. (in real life, Alan Cumming didn't want to use that painful make-up again without Bryan Singer at the helm)
 * Myers in the Hellboy movie. In the sequel, he's said to be in Antarctica.
 * The absence of Tank, who survived The Matrix, is explained away in Reloaded by Zee saying she had lost two brothers to the Nebuchadnezzar, implying that Tank had been killed.
 * Dan Cain, the nominal hero of the first two Re-Animator films, is nowhere to be seen by the time the third movie rolls around. This is Handwaved away by West remarking that his "last assistant turned state's evidence on [him]".
 * Bill Murray's Bosley is conspiciously absent (save for a photo on his family's wall) in Charlie's Angels: Full Throttle, having been replaced by his adoptive brother, played by Bernie Mac. Despite the fact that Bosley appeared to be beloved by the Angels in the first film, here they never seem to miss him for even a moment - nor is his absence (and current whereabouts) ever explained.
 * All the ship's crew from the first Pocahontas apart from John Smith.
 * Chris Klein was the only dude from the original five guys not included in the third American Pie movie, American Wedding. Word of God says they simply had nothing for him to do, so the character just wasn't included.
 * American Pie Presents: Band Camp, arguably the series' first foray into hardcore Sequelitis, had a line about why the main characters from the previous films weren't around.
 * A famous example is The Lost World: Jurassic Park. It focuses on secondary (and, in the first novel, deceased) character Ian Malcolm; as such, main characters Alan Grant and Satler are not seen or mentioned, while the kids and Hammond only cameo early on.
 * The third is focused on Grant. He's still good friends with Satler, so she appears twice in the movie (one of the scenes indirectly leads to The Cavalry's arrival). Malcolm is remembered in one scene. Hammond and inGen aren't mentioned directly (only in scenes such as when they visit an abandoned facility).
 * Short Circuit 2 didn't have Steve Guttenberg and only had a brief voice-only cameo from Ally Sheedy, despite them being the main characters of the previous film (aside from the robot, obviously). The movie instead focuses on Steve Guttenberg's Indian sidekick Ben, inexplicably played by Fisher Stevens.
 * The Austin Powers films gleefully embraced this trope in its mockery of James Bond film clichés. The love interest from the first film is retconned as always having been a fembot in the second just to get rid of her, and Heather Graham's character from the second film doesn't even get a nod in the third.
 * Galleria (Raven-Symone) in the third The Cheetah Girls "goes to college". In reality, backstage catfights between Raven-Symone and the rest of the cast led to her leaving the defictionalized band and the movie, and for Adrienne Bailon's character leaving That's So Raven.
 * Yori's rather conspicious absence from Tron: Legacy has been the source of plenty of speculation. The Betrayal comic shoves Dr. Baines-Bradley on a bus to Washington DC, but no mention at all is made of what became of her program.
 * Julia Roberts and Catherine Zeta-Jones don't appear in Ocean's 13. This is explained at the beginning of the movie with about two lines, the main one of which is "it's not their fight." However, given the nature of the motivation for the Ocean crew, this makes at least a little bit of sense.
 * Bernard the Head Elf from The Santa Clause inexplicably vanished when the third movie rolled around.
 * That's because he discovered Math.
 * Although Gracie and Eric appeared to be starting a relationship at the end of Miss Congeniality, Eric is nowhere to be seen in the sequel.
 * The live-action Transformers film series has plenty of these. For starters, there are several secondary human characters from the first film who are never seen or mentioned in either sequel. Nobody really cared about them, though, since one of the most common complaints that the hardcore fans had about all three films (and especially the first one) was that the secondary human characters were given far too much screen time.
 * As for the Transformers themselves, this was averted with the surviving Autobots from the first film, all four of whom returned for both sequels. On the Decepticon side, however, Barricade was not shown to be killed in the first film, yet despite one of the writers claiming that his fate had been intentionally left unresolved as a Sequel Hook, he was a no-show in the second film and only received a brief uncredited cameo in the third film.
 * Sideswipe and Wheelie were the only two new Autobots introduced in the second film who returned for the third film. Skids, Mudflap, Jolt, and at least one of the Arcee sisters survived the second film, but were absent for the third film. The Decepticons, however, averted this going from the second to the third film as Megatron, Starscream, and Soundwave were the three Decepticon survivors from the second film and all three returned with significant roles in the third film.
 * This even happens mid-movie in these films. Remember Miles? Neither does anybody else.
 * In the third film, Mikaela and Sam apparently broke up off-screen. This was due to Megan Fox getting fired, having insulted Michael Bay in the press.
 * The titular St. Bernard of the Beethoven movies gets given away to the brother "temporarily" (and permanently) in the third movie.
 * Wrath Of The Titans opens with Perseus standing over the grave of IO, his love interest from the first film. No really, it does.

Film- Western Animation

 * All the ship's crew from the first Pocahontas apart from John Smith.
 * Bo Peep, who was Woody's love interest in the first two Toy Story films, was given only a brief, yet sad, mention in the third, where it was indicated she had been either given away or sold off, along with a few other toys.

Literature

 * Vittoria Vetra from Angels & Demons doesn't appear in The Da Vinci Code--the last time Langdon saw her is mentioned vaguely, but after that she's just plain forgotten.
 * And Sophie Neveu didn't even get a mention in The Lost Symbol.

Live Action Television

 * The character of Miranda is missing from several of the last episodes of the Lizzie McGuire' TV show, and is only mentioned in passing as being "in Mexico with her family" during the Lizzie McGuire movie.