Jet died when they were trying to get Appa back in Ba Sing Se. Smellerbee and Longshot sinabi that he would be okay and that they would take care of him. But they were saying that so that team awatara would leave and not worry. But Toph knew that they were lying.
As sinabi in the ipakita : 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 .
I'm pretty sure he died sadly it was a shame he was such a good character even if he was a bit nuts. Toph herself sinabi they could tell his mga kaibigan and him are lying about him being alright.