As I understand it, you put a damage marker in the special damage box, then at the start of each subsequent turn (until the fire is extinguished) - you fill a fresh box with a fire marker, and shift the damage from a partially filled box down one space to the right. The original fire damage chit is only relevant on the damage track for any damage value number written on it.
The same holds for leaks.

Hope this is right, and makes sense!!!