Does fire go away after saving?
Using the saving mechanism found in the rules, all exploration tiles except the one on which every explorer is standing are discarded. Any fire tokens on these discarded cards are also discarded. While the card on which the explorers are standing is saved rather than discarded, a fire token on that card is also discarded because the mechanism does not provide a means of saving its presence.

That said, the saving mechanism provided by the rules is meant to be quick and simple to use, and nothing stops you from using a more thorough saving mechanism. For example, you could store a fire in the dice bag to indicate there was a fire present on the explorer's tile. Or you could note the detailed state of the game (perhaps in the Cartographic Notebook add-on or this printable version (.pdf, 1.2 MB)) to perfectly restore the state of the game.