Did Jet Just Die?

In the DVD release, in the Avatar Extras for "The Ember Island Players", a commentary bubble stated,

"For the record: Jet is dead."

The authors also commented that Nickelodeon wouldn't let them depict a clear death.

We never actually see him die. During his last fight Jet gets hurt by Long Feng, they hint of his death when Toph says that Smellerbee is lying about Jet being okay but that is about it. I think Nickelodeon just didn't want to say it in a children's show, because he was dead and if they say it kids might get upset but if you watch the show you see that Jet never comes back. There are also websites that say he was dead. Finding a canon answer might be hard but I think what was said in the show is enough to assume so!

Jet was hit by a pretty powerful blow, most likely causing internal damage, though it being a kid's show they couldn't show any blood coming from the mouth, which is usually what they do to show that someone is bleeding internally. I think longshot then does a mercy kill which is why he gets an arrow ready.

According to the wiki:

Jet insisted to Katara that he would be all right, but Toph sensed that he was lying. Smellerbee remained by Jet's side, crying, while Longshot notched an arrow and aimed it at the entrance of the room, ready to prevent any more attacks. Jet subsequently died from his serious internal injuries.

Also, from the same wiki page:

Bryan and Mike explained in the commentary for "The Ember Island Players" that Nickelodeon was against showing kids being fatally wounded or outright killed, especially by a violent attack. It also could not be revealed during the rest of the story, so Toph sensing that Jet was lying about how he would be okay was how they chose to confirm it for the viewers. Also, in the Avatar Extras for "The Ember Island Players", a commentary bubble stated, "For the record: Jet is dead".

In my annotated copy of The Promise, there is the following commentary:

Transcript of the commentary (which is a bit difficult to read):

GLY: The Freedom Fighters are such great characters. In an early draft, I had Jet here alongside Smellerbee, Longshot, and Sneers. I thought his fate was ambiguous enough that he could be brought back. (And Sokka agrees with me. Take a look at "The Ember Island Players.") Nope. The world of Avatar is not the Marvel universe. Here, dead means dead.

(Longshot and Sneers are there too, just cropped out because of the image size limit.)

According to earlier commentary bits, "GLY" is Gene Luen Yang, who is credited with writing the script of this comic. Along with the other answers, I think the authorial intent is pretty clear here.

If you only care for in-universe answers, Jet does not appear at any point in this comic, but he is also not mentioned by any of the characters as far as I can tell. Smellerbee and the others re-formed the Freedom Fighters without him:

I think it is implausible that they would have done so without even mentioning his name, were he still alive.

In the Legend of Korra comic named Turf Wars, there is a an opposing character that wields two curved end swords, the weapons are used the same that Jet had wielded them. The character also looked very similar, and his age seemed to match somewhere between Tenzin and Korra. Its quite possible that the character could have been Jet's son, which means he didn't die. (Just a theory)

*Spoiler Alert for Cowboy Bebop.

Since Jet is most likely an animation nod to Cowboy Bebop, (name obviously, but instead he looks like Spike, complete with "cigarette" wheat straw) it's appropriate to keep his "death" ambiguous. If you've watched the series, you'll get the really cool, clever Easter Egg.