He will take over the project from David O Russell, who parted with Sony over plans to use only the game's basic premise, Variety reports.
Burger will also write the screenplay adapted from the title, which follows Francis Drake's adventures as he seeks out the lost treasure of mythical land El Dorado.
Uncharted is reportedly being eyed as an Indiana Jones-style franchise, after the PlayStation game series sold over 6 million copies.
Mark Wahlberg signed on to the project in January, saying that he will be appearing alongside Robert De Niro and Joe Pesci.
The shoot was intended to start this summer, but it is unclear whether Wahlberg will remain in the lead under Burger.
Before his departure, Russell had reassured loyal fans of the game that he would make an adaptation that was true to its spirit.
Details of Burger's version remain under wraps, but producers reportedly "sparked to [his] take" on the film, which is said to be closer to the game.
Watch the trailer for Limitless below: