If Jon Snow indeed was the son of Lyanna and Rhaegar, how did he burn his arm in book one when he tried to save the Lord Commander Mormont from the wight? If he was part fire, how could he have singed himself?
Targaryens aren't fire proof or fire-made-flesh. That is a wrong assumption on part of a few fans who are making too much of Daenerys' ability to avoid serious burns.
But we do know that Targaryens are not fireproof, this is something that appears to be unique to Daenerys.
- Viserys III Targaryen died of being burnt by molten gold.
- Rhaenys Targaryen was burnt to death by Vhagar and Sunfyre. (We can't be sure though that if it was the fire that killed her or the fall. But we do know her corpse was so badly burnt that it could not be recognised and that she was indeed dead).
- Rhaenyra Targaryen was burnt by Sunfyre.
- Aegon II was paralyzed due to injuries given by fire of Rhaenys Targaryen's dragon Meleys the Red Queen.
- Aegon V was burnt with Prince Duncan Targaryen at Tragedy of Summerhall. That left House Targaryen family tree limited to one lonely branch of Jaehaerys II.
- Aerion Brightflame died after drinking Wildfire. I suppose we can call it internal chemical burning.
So it is established that Targaryens can burn and die. So it should not come as a surprise if Jon burnt his hand. Only Targaryen who shows immunity (relatively) to fire is Daenerys Targaryen. Would have marked this question as duplicate with my other answer How did Daenerys avoid injuries in Vaes Dothrak but since their titles are different, I'd suggest you should read that answer as well.