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.