The Cat in the Stacks Mysteries: Peter Vanderkeller, Athena College's library director (and Charlie's boss) in the first six books, abruptly sent in a resignation letter in book 7; he is later discovered to have been murdered, and the letter was sent by his killers to delay the discovery of their crime.
Empress Nerissa and her entire family die offscreen in a bloody assassination just before the opening scenes of the second book of Patricia Bray's Chronicles of Josan trilogy. Considering her importance in the first book, this comes as a bit of an immediate shock, but it isn't done wastefully or for no reason. Her death shapes the circumstances surrounding Josan's life for the entire rest of the trilogy.
Then in The Last Battle, Eustace and Jill learn that their friend from the last book, Rilian, "has been dead for over two hundred years." Presumably Puddleglum has, too. In fact, that book begins with the deaths of everyone from our world who ever visited Narnia, exceptSusan, in a train crash. But due to Narnia doubling as some kind of entrance to the afterlife, nobody actually notices they're dead until the end.
In the Dragons of Requiem prequel trilogy Dawn of Dragons, Prince Sena Seran is set up to be a main character in Requiem's Song. Two chapters into the sequel, Requiem's Hope, he hangs himself.
In the third book of The Elric Saga by Michael Moorcock, Count Smiorgan Baldhead, who had traveled with Elric in the last book, gets unceremoniously abandoned by Elric and left to his death during the battle with Melnibone. However, Elric does get called out on it. It also turns out that this is actually an inversion, since Smiorgan's death occurred in the very first Elric story ever published. However, that story was relatively far into Elric's continuity, so he got brought back later for another story set earlier. Later collections published the stories in their order in continuity, rather than the order they were originally published.
A hallmark of the Erast Fandorin novel series. If a character from a previous Fandorin book pops up in a new one, there's a pretty good chance that character is gonna die.
The first is Count Zurov, a pivotal character in Fandorin #1, The Winter Queen (at one point he saves Fandorin from being murdered by the bad guys). Zurov pops up again in Fandorin #2, The Turkish Gambit, only to be murdered by the bad guy as part of the bad guy's espionage plot against Russia.
In the first chapter of Eldest (Book 2 of the Inheritance Cycle), Ajihad is killed by Urgals, and Murtagh is presumed dead as well (although he isn't really). Also, the Razac are killed early in Brisingr.
In the first book, the character Genarro actually lives, and isn't ignominiously eaten while on the toilet like he was in the movie. He even gets to beat up a raptor. By the second book, he's died of dysentery.
Accidental inversion in that Ian Malcolm, the main character of the second book, dies at the end of the first one. He gets better, though.
An interesting case happens in the Stephen King novels Desperation and The Regulators, which are not actually sequels to each other but twinners (2 books that feature the same characters, but are set in separate fictional universes). In Desperation, Mary Jackson and David Carver are two of the only four people who make it all the way to the end of the story. In The Regulators however, they are among the first three (four if you include Hannibal the dog) victims to fall to Tak's creations before the end of the fourth chapter.
The Lord of the Rings: Boromir survives all through The Fellowship Of The Ring only to die in the very first chapter of The Two Towers after an off-screen skirmish (that arguably started at the end of the first book). Subverted in the movie adaptation, where he is killed at the end of the first movie, possibly to hype up The Climax.
The main character of Raymond E Feist's Mistress of the Empire series spends the first two books in various political machinations to secure her title and lands. The triumph of her goal is to have a son, who dies in a faked accident two pages into the third book.
Uncle Press, the uncle of Bobby Pendragon, is killed off in the second book of the The Pendragon Adventure series, so that Bobby can take his role as head Traveler in the fight against Saint Dane.
Precious dies in the first chapter of the sequel to Push, The Kid, when her son Abdul is 9. Justified Trope because Precious was diagnosed with HIV shortly after he was born.
Between books 11 and 12 of Ranger's Apprentice, Alyss and Crowley have both died, and King Duncan is on his deathbed. Also, Halt is retired.
In Redwall, Friar Hugo survives the entire first book, even being held hostage by one of the entire series most dangerous villains in the climactic battle. In the third book (chronologically, the next after the first one), he is killed off-screen in one of the first chapters.
Shadow Moon, a sequel to the movie Willow, kills off Madmartigan and Sorsha in the first chapter.
They survive the events of The Shining, but by the time Daniel Torrance has to deal with a new threat in Doctor Sleep, both Wendy Torrance and Dick Hallorann are long dead (only to be expected, since it's more than 30 years later).