Once fired they will either lose their flame and become normal arrows, or break and can't be picked up again. Occasionally they will be able to be picked back up again in their fiery form.
Any wooden arrow fired from the Molten Fury will instantly become a flaming arrow, though they will still deal the same amount of damage as a regular wooden arrow.
Since you can get a surplus of gel really early in the game, you can put it to use making flaming arrows.
Prior to version 1.0.2 wooden arrows fired from the Molten Fury did the same amount of damage as a flaming arrow.