Path of Burning Rage

Typical barbarians harbor a fury that dwells within. Their rage grants them superior strength, durability, and speed. Barbarians that walk this path instead learned to transform their rage into a mantle of primal fire that engulfs them, tapping into an inner inferno that threatens to burn all that is around them. The Path of Burning Rage is a path of untrammeled fury, slick with blood, fire, and ash.

Ignited

Starting when you choose this path at 3rd level, you glow red and heat up when you enter a rage, causing a scorching light to radiate from you, pour out of your eyes and mouth, and threaten to char you. While raging, you are immune to fire damage and gain resistance to radiant damage. Additionally, when you hit a creature with a melee weapon attack, you can deal an additional 1d6 fire damage to the target, in addition to the weapon's damage. This damage increases to 2d6 at 11th level, and 3d6 damage at 20th level.

This fire damage replaces the additional damage that is normally granted by rage.

Fuel the Flames

Beginning at 6th level, when you reduce a hostile creature to 0 hit points while raging, your rage lasts an additional round and all melee attacks that you hit before that rage ends do an additional 1d6 fire damage. This effect stacks, to a maximum of 5d6 extra fire damage on hit and a maximum of 5 extra rounds of raging.

When your rage ends, you heal an amount equal to a roll of the d6 die you've accumulated from this trait, but cannot regain hit points any other way until you take a short or long rest as your skin burns and smokes. A Greater Restoration spell or similar magic could end this effect, allowing you to heal sooner. 

Literal Fire

Beginning at 10th level, your arms and legs ignite completely when you rage, as your skin chars and cracks. When you enter rage, you take 2d8+4 necrotic damage that cannot be reduced or negated in any way. However, whenever a creature hits you with a melee attack while you're raging, they take 2d6 fire damage or radiant damage (your choice). You can deal 2d6 fire damage or radiant damage to any creature grappled by you at the start of their turn if you are raging. 

Additionally, you know the Produce Flame cantrip. Your Spellcasting modifier for it is 8 + your proficiency bonus + your Constitution modifier. You can cast this cantrip while raging.

Starfire Havoc

Starting at 14th level, you learn to channel your fiery rage into a single attack. As an action on your turn, you can expend 3 uses of rage to deliver a devastating punch, making a melee attack. On hit, the target takes 17d6 + your Constitution Modifier of fire damage or radiant damage (your choice) and a devastating explosion erupts from your fist. Each creature in a 30-foot radius Sphere centered on the target must make a Dexterity saving throw (DC equals 8 + your proficiency bonus + your Constitution modifier). A target takes 10d6 fire damage on a failed save, or half as much damage on a successful one. This explosion ignites any flammable objects that aren't being worn or carried. 

Using this attack ends your current rage, if any, and grants you 2 levels of exhaustion even if you miss the attack.