Lit Bronze fire arrows are used during the Underground Pass quest. They are created by using a damp cloth on regular bronze arrows to create unlit bronze fire arrows, which are then lit by using them on a fire. Any lit fire arrow can be used to pierce the rope in the Underground Pass Dungeon, which lowers a bridge. This is necessary to complete the quest.
Bronze fire arrows have the same Ranged Strength bonus as regular bronze arrows, and the fire provides no additional benefit.