Indiana Jones and the Temple of Doom
The Karate Kid
Star Trek III: The Search for Spock
Revenge of the Nerds
Romancing the Stone
Lots more, obviously, but those are stand-outs. Not all in terms of quality, obviously -- Red Dawn is trash, no matter how you slice it -- but it's interesting to note that of the ten movies I list here, all of them among that year's twenty top-grossing films, two have been remade. (So has a third of the 1984 top twenty, actually -- Footloose!)
Meanwhile, Casablanca has never been remade. Its story beats have been mined a lot, and the film has an enduring place in popular and wider culture, but no one has yet actually made a new version of the film. Will it happen? Who knows? Anyhow, all hail the newest Old Movies!