It is obtained from the Giant Obsidian Vault in the Mushroom Vault of a Troll Cave, below the Cloud Cottage, near the Thornlands. It allows the player to access the Thornlands, by burning Thorns into Burnt Thorns. Right-clicking a vine block will burn that individual block. Holding then releasing a right-click on air will burn a 4x4x4 patch, centered on the player. Once burned, thorns will disintegrate when punched.
Acquiring the Lamp of Cinders gives the Achievement, I Wish For More Burning.