No, Flash Fire does not siphon off attacks Lightning Rod and Flash Fire work slightly differently.
Lightning Rod will siphon any single target Electric type attack to the Pokemon with the ability. That attack will then be negated, raising that Pokemon's Special Attack by 1 stage.
Flash Fire does not siphon off Fire type attacks to the Pokemon with the ability. Rather, if the Pokemon with the ability is hit with a Fire type attack, of any source, the ability will trigger. Instead of raising the Pokemon's Attack by 1 stage, it simply increases the damage that Pokemon deals by using Fire type attacks for as long as it stays in battle. Instead of being a "scaling" increase to Special damage output, it gives a flat +50% to Fire type moves.
To the best of my knowledge, there is no Ability that siphons off Fire type attacks