In the books, Dany is only immune to fire for that brief period it takes to hatch the dragons' eggs. According to GRRM, it was the combination of the fire and the sacrifice of a life (Mirri Maz Duur) that allowed Dany to briefly become immune. It's not an ongoing thing. However, that doesn't mean that Targaryens cannot be more resistant to fire and heat than other people, but it doesn't come up (that I remember) in the books.

In the TV show though, they seem to be saying that Dany has a radically enhanced resistance to heat at all times. Viserys might have had it as well, or it might be a reference to a kind of 'true dragon' gene that only certain people have (like Dany and possibly Rhaegar and their father). I think it's a way of setting up later events a bit more ahead of time.