In Game of Thrones S08E03, "The Long Night", Daenerys's dragon Drogon blasts fire at Jon Snow to burn the wights surrounding him and he is unscathed, being half Targaryen.
But later the White Walker dragon Viserion tries to kill him with blue fire and he hides, as if he could be burnt. Is he just scared?
No, Jon is not immune to fire. Remember in the first season, when a wight attacks Castle Black and tries to kill Lord Commander Mormont. Jon throws a lantern at it, and burns his hand in the process.
Jon tried to shout, but his voice was gone. Staggering to his feet, he kicked the arm away and snatched the lamp from the Old Bear's fingers. The flame flickered and almost died. "Burn!" the raven cawed. "Burn, burn, burn!"
Spinning, Jon saw the drapes he'd ripped from the window. He flung the lamp into the puddled cloth with both hands. Metal crunched, glass shattered, oil spewed, and the hangings went up in a great whoosh of flame. The heat of it on his face was sweeter than any kiss Jon had ever known. "Ghost!" he shouted.
The direwolf wrenched free and came to him as the wight struggled to rise, dark snakes spilling from the great wound in its belly. 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
Additionally, Targaryens were never meant to be immune to fire allways:
Lastly, some fans are reading too much into the scene in GAME OF THRONES where the dragons are born -- which is to say, it was never the case that all Targaryens are immune to all fire at all times.
So Spake Martin - Egg and the Targaryens
Stone does not burn, Harren had boasted, but his castle was not made of stone alone. Wood and wool, hemp and straw, bread and salted beef and grain, all took fire. Nor were Harren's ironmen made of stone. Smoking, screaming, shrouded in flames, they ran across the yards and tumbled from the wallwalks to die upon the ground below. And even stone will crack and melt if a fire is hot enough. Harrenhal could still be attributed to dragonfire being hot, as opposed to magical.
From an interview with GRRM.
Granny: Do Targaryens become immune to fire once they "bond" to their dragons?
George_RR_Martin: Granny, thanks for asking that. It gives me a chance to clear up a common misconception. TARGARYENS ARE NOT IMMUNE TO FIRE! The birth of Dany's dragons was unique, magical, wonderous, a miracle. She is called The Unburnt because she walked into the flames and lived. But her brother sure as hell wasn't immune to that molten gold.
Revanshe: So she won't be able to do it again?
George_RR_Martin: Probably not.
As to whether Daenarys is actually fire-proof there's some very interesting reading over on Reddit. Some of this explains that in-universe she is not fire-proof.
Dany hit him. “No,” she screamed, swinging the lash with all the strength that she had in her. The dragon jerked his head back. “No,” she screamed again. “NO!” The barbs raked along his snout. Drogon rose, his wings covering her in shadow. Dany swung the lash at his scaled belly, back and forth until her arm began to ache. His long serpentine neck bent like an archer’s bow. With a hisssssss, he spat black fire down at her. Dany darted underneath the flames, swinging the whip and shouting, “No, no, no. Get DOWN!” His answering roar was full of fear and fury, full of pain. His wings beat once, twice [...]
Her skin was pink and tender, and a pale milky fluid was leaking from her cracked palms, but her burns were healing.
A Dance With Dragons, Page 699 Google Books
As shown by her inner dialogue, and the later burns healing Daenarys is clearly not completely fire-proof.
Are you sure that he actually got hit by the fire?
Ignoring all questions of the TV vs Book rules of whether Targaryens are consistently immune to fire, we saw in season 1 when Daenerys entered the fire and survived, her clothes did not. Even if she is immune to heat because of her Targaryen blood, fabric is not.
Even if Jon is in theory immune to fire like Dany, had he been hit directly by the dragon fire we would have seen some fire damage to his armour and clothing, but we didn't.
