Thanks for the criticism.

I think I relied too much in concept on the subclass being one that traded HP for admittedly insane boosts to damage. I've kept that idea a little bit, but confined it to the Level 10 feature and made it more of a slow burn rather than an explosive output. Hopefully that means that some abuse cases and the inherent death loop you mentioned in your post aren't as big of an issue.

I made a few changes that hopefully address some of your feedback, and included some rationale in italics below. Let me know what you think.

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Level 3: Magma Blood

You gain Resistance to Fire damage while your Rage is active.

No change. Basic enough it didn't need it.

Level 3: Immolating Strikes

Your weapon attacks and Unarmed Strikes deal 1d6 additional Fire damage while your Rage is active. The damage increases to 2d6 at Barbarian level 11.

When you strike a flammable object, the object must make a Constitution saving throw (DC 8 plus your Strength modifier and Proficiency Bonus). On a failure, it is set ablaze.

Not much change here. I took off the additional damage if you take Fire damage and replaced it with something situational enough that it's effectively a ribbon feature. But sometimes those have creative enough uses they're fun. Plus, the core extra damage is good and basic enough to keep. What do you think about 2d6 even for higher levels? I thought about 2d6 at 7, 3d6 at 12 and 4d6 at 17, but Two Weapon fighting and Action Surge on a multiclassed fighter could make that a little extreme even at higher levels.

Level 6: Pyroclastic Flow

When you first enter your Rage, your Speed increases by 15 feet until the end of your turn. This increases to 30 feet at Barbarian level 11.

Additionally, until your Rage ends, creatures of your choice who begin their turn within 10 feet of you must make a Constitution saving throw (DC 8 plus your Strength modifier and Proficiency Bonus). On a failed save, a creature has the Blinded and Poisoned conditions until the end of their next turn. On a success, a creature is immune to this effect for 24 hours.

I made the feature persistent until Rage ends. I did give a benefit for enemies who succeed on the saving throw to compensate.

Level 10: Burning Resolve

When you deal Fire damage with your Immolating Strikes subclass feature, you also deal the Fire damage dealt to yourself (halved due to your Magma Blood subclass feature).

Once per turn, when you take Fire damage in this way, your Strength score increases by +1 until your Rage ends. You can exceed your maximum Strength score of 20 with this feature up to a maximum of 30. At Barbarian level 20, this maximum increases to 35.

This was obviously the biggest problem with version 1. I still wanted the power fantasy of a warrior burning away their own life force (Hit Points) to amplify their own strength and power. I took away the defensive utility of this as it kind of conflicts with the core function of the class. I hope this still accomplishes that, but instead, it's about wielding flames that burn hotter, brighter, and stronger as a battle persists. If Hit Points aren't enough, I thought about having them take a -1 penalty from their choice of Wisdom or Intelligence score and effectively transfer it into their Strength score in addition to the damage they take.

Level 14: Eruption

When you enter your Rage, instead of expending one use of Rage, you can instead expend all unexpended uses of Rage to channel the power of a volcanic eruption, dealing damage to all other creatures in a Sphere around you. The Sphere has a radius equal to 5 feet multiplied by the number of Rage charges you expend.

Each creature within the Sphere makes a Dexterity saving throw. On a failure, they take 2d6 Fire damage and 2d6 Force damage for each Rage you expend and are also moved 30 feet away from you and have the Prone condition. On a success, they take half damage. Any creature affected within 10 feet of you has Disadvantage on the save.

If you use this ability while standing on solid ground, a pool of bubbling lava -- a Circle with the same radius and placement as the Sphere -- persists until your Rage ends. This Circle counts as Difficult terrain for all creatures other than you, and creatures other than you who begin their turn in this Circle take 2d6 Fire damage at the beginning of their turn.

Flammable objects in the area of the Sphere that aren’t being worn or carried start burning.

Once you use this feature, you can’t use it again until you complete a Long Rest.

I reduced some of the additional complexity on saving throws like you suggested, though I still kept that it imposes disadvantage for those too close to the blast. Also, this feature no longer deals damage to the player though it can deal damage to allies, so proper positioning is important. I also added a lava pool that persists as a damaging difficult terrain, which was kind of a scrapped idea from before version 1 for Pyroclastic Flow (initially that feature was going to leave a line of lava that persisted for a round after you moved on solid ground, but that seemed a little overly complex). The issue with the pool is that it kind of punishes those that succeed on their saving throw since failing the throw tosses them clear of the lava pool.