It’s clear that he’s dead, though what happened to him is not revealed until the sequel game Rise of the Tomb Raider.
The trip to Yamati in the game is undertaken as a simple archeological expedition based on Lara’s theory of where a lost civilization may be found.
In the game, Lara’s father is only mentioned in conversation, he is not a character himself.
The inciting incident of the new Tomb Raider movie is the fact that Lara Croft discovers what her father was doing just before he disappeared, leading her to follow in his footsteps and head for the island of Yamati.