The Targeryens (I assume that they are who you mean by 'Dragonborn'), are not immune to fire as stated multiple times by George RR Martin.
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.
From a chat...
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.
In response to the followup question asking if Dany would survive the fire again.
So a dragon can burn, bite, scratch, or sit on a Targeryen to kill them.