There was a scene in the original screenplay that addressed this. Luke sold it to a 'used speeder dealer'. Beyond this, its fate is unknown.
EXT. TATOOINE - MOS EISLEY - STREET - ALLEYWAY - USED SPEEDER LOT.
Ben and Luke are standing in a sleazy used speeder lot, talking with a tall, grotesque, insect-like used speeder dealer. Strange exotic bodies and spindly-legged beasts pass by as the insect concludes the sale by giving Luke some coins.
LUKE: He says it's the best he can do. Since the XP-38 came out, they're just not in demand.
BEN: It will be enough.
Ben and Luke leave the speeder lot and walk down the dusty
alleyway past a small robot herding a bunch of anteater-like
creatures. Luke turns and gives one last forlorn look at his
faithful speeder as he rounds a corner.
Some of Luke and Ben's dialogue made it into the film, but not the used
car speeder dealership, which was deleted, presumably because it adds little to the plot.
hat tip to Megaptera novaeangliae for the reminder.
We learn in the junior novelisation that he sold his speeder for 2000 Imperial credits (described in the original novelisation as "a number of small metal shapes").
Facing the speeder dealer, Luke said, “All right, give it to me, I’ll
take it.” The speeder dealer, an insectoid alien, had finally agreed
to buy Luke’s cherished landspeeder for two thousand credits.
The scene made it into the Marvel comic adaptation, issue #2
We have some additional info about the creature that purchased the speeder, a vuvrian called Wioslea. The company that she worked for was called Spaceport Speeders.
The negotiations were captured in the Star Wars Radio Play.
Luke Skywalker: I want to sell this speeder.
Obi-Wan Kenobi: If we can get a fair price.
Luke Skywalker: Yeah. I paid twenty-four hundred for it, and I put a lot of work into it. I think it's worth a lot more.
Wioslea: Fifteen hundred is all I can offer you, human.
Luke Skywalker: Fifteen hundred? Look, this landspeeder's in great shape, even if it's not very pretty.
Obi-Wan Kenobi: We can accept no less than two thousand.
Wioslea: Your grip on reality is fragile, humans. It's the newer models that are in demand now, not these old ones. I shall probably have to sell it at a loss.
Obi-Wan Kenobi: But it's still not a fair price.
Wioslea: You touch a responsive chord in my… charitable nature, humans. Sixteen hundred.
And it was confirmed in the EU novel Star Wars Journal: The Fight for Justice that Obi-Wan used a Jedi mind-trick to convince Wioslea to pay precisely what the Jedi needed to pay for passage off the planet.
We left Solo and Chewie at the cantina and headed over to a used
speeder lot with Artoo and Threepio. As usual, the dealer tried to cut
us down in price, and Ben's use of the Force saved us again. We needed
two thousand for Captain Solo, and that's what Ben got us. I'm sure
the dealer's going to wonder why he was so generous when he recovers,
but he can still sell the landspeeder at a profit, so we didn't feel
The speeder ended up being purchased from the used speeder lot and used by a racer called Wade Vox in the game Star Wars: Demolition.
We have confirmation from Star Wars Insider #50 that this was indeed Luke's speeder.
At the end of the game Wade is killed and the fate of the speeder is unknown, but presumably it ends up in the possession of Jabba the Hutt.