We've seen his aunt survive all fire and I even remember she almost enjoyed the scalding bath.
Jon was resurrected by the fire god. Is Jon now immune to fire, and just doesn't know it?
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As far as the books go, GRRM has said that even Daenerys is not immune to fire and surviving the pyre was a one-time thing related to the dragons.
In the show, Daenerys appears to be permanently immune, however we actually see Jon burned by the lamp that he uses to burn the wight in the Lord Commander's quarters -
So it would appear that he is not immune to fire as Daenerys is.
Edit: As for being a fire wight, we don't know much about what abilities this potentially comes with. We do know that Beric Dondarrion uses blood to light his sword, as seen when he fights the Hound. This may be because he is a fire wight, or he may be using some sort of blood magic.
It could be easy to assume that a fire wight would have fire immunity - after all, the definition given by GRRM is "a wight animated by fire," as opposed to ice. However this has not been confirmed in the show or in the books.
If being a fire wight does give you fire immunity, it could even be possible that Daenerys is a fire wight. After all, if she is not immune to fire because she is a Targaryen, why is she? This is all just speculation, so until they actually show Jon or Beric consumed by flame and left unscathed we really don't know.
Jon was burned by the fire he used to kill the wights in Season 1 that invaded Castle Black.
Quotes from the book, which since this is Season 1, the scenes in the show remain highly faithful to. The burns were quite severe and took weeks or months to heal properly.
Jon plunged his hand into the flames, grabbed a fistful of the burning drapes, and whipped them at the dead man. Let it burn, he prayed as the cloth smothered the corpse, gods, please, please, let it burn.
A Game of Thrones - Jon VII
"You do not look well. How is your hand?"
"Healing." Jon flexed his bandaged fingers to show him. He had burned himself more badly than he knew throwing the flaming drapes, and his right hand was swathed in silk halfway to the elbow. At the time he'd felt nothing; the agony had come after. His cracked red skin oozed fluid, and fearsome blood blisters rose between his fingers, big as roaches. "The maester says I'll have scars, but otherwise the hand should be as good as it was before."
A Game of Thrones - Jon VIII
He flexed his burned hand as he rode, opening and closing the scarred fingers. They still pained him, but it felt good to have the wrappings off.
A Game of Thrones - Jon IX
He gave Jon a long, searching look. "But we were speaking of hands. How is yours?"
"Better." Jon peeled off his moleskin glove and showed him. Scars covered his arm halfway to the elbow, and the mottled pink flesh still felt tight and tender, but it was healing. "It itches, though. Maester Aemon says that's good. He gave me a salve to take with me when we ride."
A Clash of Kings - Jon I
Only Daenerys has shown any kind of exceptional immunity to fire, while other Targaryens such as Viserys were not immune and died from fire/heat, suggesting this ability is not a Targaryen trait but specific to Daenerys.
Jon has been burned before when he fought off the first wight attack as Castle Black. He grabbed a lantern the wrong way and he appeared to recoil in pain as he did so. However, this was before he was resurrected and became a fire wight. We have no idea whether fire wights have any special immunity to fire.