Living Flame is a legendary crafting material used by Blacksmith Haedrig Eamon to craft a legendary 2-handed weapon axe named Cinder Switch.
Living Flame has a chance to drop off the morlu caster Sao’Thall (Embodiment of Hatred). He is a Bounty target in Act IV: The Silver Spire Level 2 — however, he spawns in non-bounty maps too. I just killed him during Bounty: Aspect of Terror, for example.
Living Flame also drops from Rhau’Kye in Act IV: The Silver Spire Level 2 and Malfeas the Abhorrent (rare spawn) in Act IV: The Silver Spire Level 2 found at the Radiant Chapel dungeon.
The Flesh Render is required to craft this weapon. Flesh Render is a random drop from Weapon Racks in the Garden of Hope Tier 1. Maybe other areas of the High Heavens.