Targaryens aren't fire-proof.
From an interview with GRRM.
Granny: Do Targaryens become immune to fire once they "bond" to their
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.