At the end of Jumanji the board game is dumped, but is later shown to show up on a beach somewhere. Is the game ever recovered at all? Does anyone else play it?
And as for the original short story, it appears to be a yes and no - Zathura is the sequel, and according to the summaries on Wikipedia, the two kids who find Jumanji at the end of the Jumanji book don't actually get around to playing it.
They pull out the board that was underneath Jumanji, a game called Zathura, which is set in space. At the end of Zathura, there's a bit of time travel that allows one of the boys to throw the Jumanji box into the garbage before they start playing it. Unless it's dug out of a landfill, I think it's supposed to imply that no one is going to play those games again.
Yes, but it's only played again after it's transformed into a video game
The 2017 sequel film Jumanji: Welcome to the Jungle shows that the Jumanji board game was indeed found on the beach, as we see at the end of the first film. From Wikipedia:
In 1996, in Brantford, New Hampshire, teenager Alex Vreeke receives the Jumanji board game, found by his father on a beach. When Alex dismisses it, the game transforms into a video game cartridge. When he plays it, he is sucked into the game.
Is this a continuation of the first film or a new continuity?
It is indeed a continuation of the first film, which you can easily tell from the opening of the extended trailer. The board game Jumanji is found on a beach, with the same appearance as in the first movie. Later, Alex reads the intro to the video game version, and it has the same poem that the board game had.
A game for those who seek to find
a way to leave their world behind.
"Who plays board games anymore?" Those of us at Board & Card Games Stack Exchange would like to have a word with whomever wrote that line!
Prior to sequel Jumanji: Welcome to the Jungle (2017), the ending of the original Jumanji (1996) film heavily implies the game is found on the beach by two young children.
As seen in this video clip starting at 3:41 the two younger children are approaching and the drums start beating. However, the final outcome is left open as the screen cuts to black.
The new sequel effectively "retcons" this as the trailer notes "for two decades it went untouched". I say retcons because while the children may not have picked up the game the beach at the end of the 1996 film is clearly not the same as the start of the 2017 film.