Both in the show and in the books, Daenerys survives Drogo's funeral pyre. Many fans mistakenly believe this is because she, as a "dragon"/Targaryen is fireproof. The TV series re-enforces this view by adding a scene in Season 1 where she handles a red-hot dragon egg without getting burned, which didn't happen in the books.
Targaryens, however, are not fireproof. They are not literally "dragons". They are HUMANS.
"You Westerosi' are all the same. You sew some beast upon a scrap of silk, and suddenly you are all lions or dragons or eagles. Such folly." Illyrio, (A Dance with Dragons)
In the history of Westeros, many Targaryens died because of fire. Rhaenyra Targaryen was roasted and then eaten by her brother Aegon II's dragon. Aerion Brightflame drank wildfire and burned to ashes. Aegon V and his son Prince Duncan both burned to death at Summerhall.
And, most importantly, Targaryens do not bury their dead. They BURN them.
On Westeros.org, on the "So spake Martin" section, in one of his MANY interviews, the author states that Daenerys survived Drogo's pyre not due to her being "fire-proof" (she's not). She survived due to the blood-magic of sacrificing Mirri Maz Durr which also hatched her dragon eggs.
However, many fanatical "Dany Fans" refuse to believe this, believing that their own ideas should take precedent over the guy who, ya know, created her. Luckily, these zealots are few and far between.
Hope this clears things up with some people. Daenerys and her Targaryen ancestors might be fire-resistant to some extent, but completely fire-proof? Certainly not.