Rabbi Yochanan said: One is liable on the damage caused by his fire on account of it being “his arrows” (it is as if he shot out an arrow which caused damage).
The Nimukei Yosef explains that this is why one is permitted to light candles Friday afternoon even though they will be burning on Shabbos; since the candles were lit from before Shabbos, which is when he shot the arrow.
The Minchas Chinuch explains further: According to Rabbi Yochanan, he is liable for the moment that he set the fire ablaze. Just as one who shoots an arrow is liable for the shooting of the arrow even though the damage which occurs afterwards is now unavoidable; so too it is with respect to one who lights a fire. Accordingly, a halachah would emerge as follows: If one lit a fire and before it had a chance to do damage he died, the inheritors would be obligated to pay (from the properties of the deceased), since the reason for liability was already completed while the lighter was still alive. This is only true if “his arrows” did not cease (when there was a fence preventing the fire from spreading, and the fence fell down after he died). However, if “his arrows” ceased before he died, they will be exempt from liability. As long as the heirs did not know about the fire, they would not be liable, for if they did know about it, and they could have prevented the fire from damaging, they will be liable, for it is their property that is damaging.