There is a novel, As Time Goes By, that relates the further adventures of the characters in Casablanca, and gives Richard Blaine some background.
I’ve not been able to finish the damn thing. 
Written by Michael Walsh, theatre critic for Time magazine, who also wrote a damn fine biography of Andrew Lloyd Webber, and later updated it.
Gone With The Wind had updates with Scarlett, Rhett Butler’s People and The Wind Done Gone. Don’t bother reading them.
Randolph and Mortimer Duke from Trading Places were given a huge chunk of money by Prince Akeem in Coming To America while living on the streets. “Mortimer! We’re baaaack!”
Indeed, the finest example of an inter-media “*wrap-up” (*from print to broadcast), is the tastefully executed Rapture storyline arc which was developed in the Bible and successfully concluded in two separate television shows (oneand two).
The story of Prince John from The Lion Winter is continued in the novel Myself as Witness by James Goldman (who wrote the play and screenplay of TLiW). Surprising considering that John is an unlikeable character in TLiW and pretty much condemned historically, Goldman’s portrait of him in the novel is actually very sympathetic. It covers events leading up to the signing of Magna Charta and John’s death the next year.
Goldman also wrote the screenplay for Robin and Marian, in which Richard the Lionhearted isn’t the main character obviously but is an important character and his death a major event in the first quarter of the movie. Richard had previously been a major character in TLiW.
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Speaking of English history, the miniseries Elizabeth R is a direct continuation of The Six Wives of Henry VIII. Keith Michell doesn’t appear in E-R (it begins soon after his death) but many other characters (Abp. Cranmer, Katherine Parr, the brothers Seymour, etc.) are played by the same actors.
There’s a line or two in Silence of the Lambs that Will Graham, the protagonist of Red Dragon, is a hopeless alcoholic in Florida now.
Frederick Forsyth wrote a sequel to Phantom of the Opera called Phantom of Manhattan. DO NOT READ THIS BOOK!!!
The Iliad never goes into what happens at the end of the Trojan War, being the story of Achilles and his own issues. Virgil picks it up again in Book 2 of The Aeneid, with the Trojan Horse and the fall of Troy.
(I assume there are other stories that cover it, but those are the only ones I remember from Classics - I know the Trojan Horse was known about by Virgil’s time, which is why he added it. No doubt some more knowledgeable Dopers will pop up to fix the glaring errors I’ve just made :))
And from about the same time, we find out what happened to Captain Nemo from 20,000 Leagues Beneath the Sea in The Mysterious Island.
Ooo, I’ve just remembered that Heimskringla concludes the story of Troy - basically, Odin escaped the fall of Troy and rocked up to Scandinavia. That’s more about the craze of tracing ancestors back to Troy, though, rather than a desire to finish the story (which I guess the Aeneid was in part as well).
Henry VI, Part 3 is all wrapped up in Richard III.
The Trojan Horse is certainly mentioned in The Odyssey. IIRC, it’s talked about in the Iliad, as well, but I can’t ecall any citations. In fact, the Odyssey is itself a continuation of The Iliad, as were a lot of lost poems, like the Returns, the Little Ilad, and others. A direct continuation of the events of the TRojan War after the Iliad is Quintus Smyrnaeus’ aptly-named PostHomerica. (If Matt Groening is smart, he can use that for his next cartoon after The Simpsons ends.)
Checked a few sources, and I confirmed my recollection that while it is mentioned in The Odyssey, it is not mentioned in The Iliad, which ends before the fall of Troy. Thousands upon millions of kids reading the Iliad for the first time have been disappointed - I know I was.
The Thursday Next novels by Jasper fforde have many examples of what happens to characters after their novels end. Miss Havisham, for instance, is a kind of the secret agent, and the Little Prince shows up at parties showing people a drawing of a hat. 