Jet died when they were trying to get Appa back in Ba Sing Se. Smellerbee and Longshot sagte that he would be okay and that they would take care of him. But they were saying that so that team Avatar would leave and not worry. But Toph knew that they were lying.
As sagte in the Zeigen : it's rather unclear lol He's thought to be dead because he was left while severely Injured. He told katara he'd be alright but toph could tell he was lying. So he is thought to be dead but know one really knows for sure .