No,Jon Snow is not immune to fire. In season 1, episode 8 he fought with a wight to protect Lord Commander Mormont. He threw a burning lamp at the wight and in the process,his hands was burned with fire.
Once the Unsullied are under her command, Kraznys angrily comments that Drogon does not obey him, to which she responds, in Valyrian, that he does not obey because dragons are not slaves. Daenerys marches forward with her army and dragons.