Outside of a Firestarter proc, there isn't a reason to cast Fire III unless you have max MP and is transitioning from Umbral Ice 3 to Astral Fire 3.

Fire III has the highest potency of any of a BLM's fire spells but because it is used to acquire the Astral Fire 3 buff, the damage it does coming out of Umbral Ice 3 is even less than Fire I. A Firestarter proc gives a BLM a reason to cast Fire III WITH the Astral Fire 3 buff in effect. Moreover, it is instant cast and costs no MP.