Believe it or not, the 1986 John Woo movie, A Better Tomorrow, the movie that kickstarted the Heroic Bloodshed genre, was actually a remake of a 1967 Cantonese movie called Ying Xiong Ben Se or "Story of a Discharged Prisoner." Tsui Hark had been toying with the idea since his days in the TV business, but because of an overwhelming workload, Hark had to pass the directorial reins to Woo.
The play The Front Page by Hecht and MacArthur was first adapted into a movie in 1931, starring Adolphe Menjou and Pat O'Brien. Howard Hawks remade it in 1940 as His Girl Friday with Cary Grant and Rosalind Russel, gender-flipping the role of reporter Hildy Johnson. In 1974 Billy Wilder flipped Hildy Johnson's gender back to male in his screen adaptation with Jack Lemmon and Walter Matthau. Then in 1988 Ted Kotcheff remade it for the satellite television age with Burt Reynolds and Kathleen Turner, flipping Hildy back to female.
Heaven Can Wait (1978) was a remake of the 1941 film Here Comes Mr. Jordan, and itself was remade in 2001 as Down to Earth. It has nothing to do with the 1943 film Heaven Can Wait (directed by Ernst Lubitsch).
Mighty Joe Young is also a friendlier remake of King Kong. Incidently, that movie was also remade in the 90's.
An unusual triple threat: Leo McCarey's Love Affair (1939), starring Irene Dunne and Charles Boyer, was remade (by Leo McCarey) as An Affair to Remember (1957) with Deborah Kerr and Cary Grant, which was remade again (by a different director) as Love Affair (1994) with Annette Bening and Warren Beatty. Ironically, An Affair to Remember is regarded as the best of the three, which has led to a general assumption that An Affair to Rememberwas the original.
Disney was to have released a motion-capture remake of Yellow Submarine in 2012 with Robert Zemeckis directing it, but it was scuttled after going over budget and Zemeckis' 2011 film Mars Needs Moms tanking at the box office.
The Birdcage, a remake of the French la Cage Aux Folles
The 2010 remake of the 2007 movie Death At a Funeral. Bizarrely unneeded, as the original came out not even three years before the film was made, has one of the same actors returning for the same character, and is exactly the same in every way except the characters are black.
The Debt, a 2011 remake of the 2007 Israeli film by the same name.
The Departed, a remake of the Hong Kong movie Infernal Affairs
Down And Out In Beverly Hills was a remake of a French farce which translated is Boudu Saved From Drowning.
Father's Day, the 1997 remake of the French film Les Compères
American Idol and its counterpart Canadian Idol are actually the US and Canadian versions of the hit British talent show, Pop Idol. This applies to any other show with the title *Insert Nationality Here* Idol.
A good chunk of Hispanic Soap Operas are either remakes of previous soaps (TV or radio) or adaptations of famous romantic books. One example who combine both is the famed "Corazón Salvaje" (title translates to Savage Heart), who began as a romantic novel, then was adapted as a soap in The Sixties, then later as a movie in The Seventies, and then again as a soap in The Nineties who unusually for the trope was claimed as the better version of them all.
Sadly, they remade it again in 2009, and ruined it again.
Game shows do this all the time. The best ones came during the '70s and '80s, where The Match Game became Match Game '7x, The Price Is Right evolved into The (New) Price Is Right, Pyramid kept climbing in dollar amounts, and Password would become Password Plus and later Super Password. There were numerous "Same shows, new hosts" examples as well, such as Family Feud (Richard Dawson, then Ray Combs) and Card Sharks (Jim Perry, then Bob Eubanks). More recent revivals tend to fall a bit flat by comparison (Match Game '98 with Michael Burger, Card Sharks '01...just, Card Sharks '01, Family Feud with pretty much everyone since Ray Combs — even an aged Richard Dawson — until John O'Hurley came along, and Pyramid with Donny Osmond).
On the other hand, GSN's remake of Lingo (hosted by Chuck Woolery) was actually superior to the original.
Same show, different name: Shenanigans (1960, local Los Angeles TV) became Video Village (1960) then became Shenanigans again (1964), P.D.Q. (1965) became Baffle (1973), Second Chance (1977) became Press Your Luck (1983), and Shoot For The Stars (1977) became Double Talk (1986).
Airplane!!: The plot and much of the "straight" dialogue were taken from Zero Hour!. Here's the dialog script, so you can see for yourself. The lines that also appear in Airplane! are in boldface.
Don Bluth's 1997 Anastasia is officially a Disneyfication/fantasticization of the 1956 Ingrid Bergman film (itself a play adaptation).
Fox specifically presented him with a list of works they owned the rights to that he could adapt. It boiled down to this or My Fair Lady.
The novel Beau Geste was adapted to film in 1926, 1939 and 1966. In 1977 a parody titled The Last Remake of Beau Geste was made. The title became not entirely true, because BBC made a television version in 1982.
The first episode of Calvin & Hobbes: The Series became another Old Shame for the author when he was older, and when he was contacted for use of it in a fanfiction-based play, he decided to rewrite it entirely and release it to the public in Season 3's bonus chapter.
Bally's classic electro-mechanical Fireball pinball was remade as a solid-state game fourteen years later as Fireball Classic, with revised rules, updated artwork, and changes to some of the game hardware.
Last Man Standing is a remake of a remake, being a remake of a Fistful of Dollars which was a remake of Yojimbo
My Little Unicorn: Starfleet Magic is an ongoing remastered version of My Little Unicorn: Believing Is Magic. Beside minor changes regarding obvious narrative flaws, not much has really changed at all.