Lit Mithril fire arrows are used during the Underground Pass quest. They are created by using a damp cloth on regular mithril arrows to create unlit mithril 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.
Mithril fire arrows have the same Ranged Strength bonus as regular mithril arrows, and the fire provides no additional benefit.