Dany's funeral pyre excursion was a one-time thing, fueled by blood magic and who knows what else, not because she herself was ever really immune to fire as a constant thing.
Several Targaryens have come up against fire or extreme heat and lost. Most recently, Viserys took a pot of molten gold to the head and died. "Big deal," you say, "it only means that he wasn't a real dragon!"
Which is why I also point out Aegon V, the dragon who hatched at Whitewalls, who died in a fire at Summerhall, along with his oldest son, Duncan the Small, aka The Prince of Dragonflies, aka Jenny of Oldstones' squeeze. Aegon V's own brother, Aerion Brightflame, thought it'd be a good idea to drink wildfire. (Spoiler: It did not work out.)
And of course we have all the poor side-branch Targs who died during the Dance of the Dragons, at least some of whom surely died thanks to dragon fire. Rhaenyra Targaryen was fed to a dragon; over-under on whether it roasted her first?
As for Dany, just Drogon's breath was enough to blister her skin in ADWD. If he'd wanted to incinerate her, he could have. You'll also notice that she treats the situation with the appropriate level of terror and does not once during it think, "It's cool, I'm immune to fire!"
Along these same lines, I don't think that Targaryens are immune to sickness, either. Someone should have told Dany that her great-great-great-grandfather Daeron II and a bunch of distant cousins died in the Great Spring Sickness. We also have alcoholic Targs (Aegon V's brother Daeron), Targs who died in war (Daeron I, Maekar I), Targs who were allegedly stabbed to death by the Iron Throne (Maegor I), Targs who just took a bad hit to the head (Baelor Breakspear), and those who were supposedly poisoned (Baelor I).
The lessons to take away from this:
1. Targaryens are not immune to fire and probably not immune to any other calamity.
2. Saying you're "blood of the dragon" and super-duper special does not make it so.
3. Inbreed enough to preserve exotic features and you can get the Westerosi to believe anything.
And if that isn't enough, read this and this.
Remember, only you can prevent people from believing that a Targ will survive a forest fire.
Edited by Apple Martini, 17 February 2012 - 02:14 AM.