There is a lot about burn here on the forums, and I wanted to use this as a jumping off point for discussion.
I want to put this out in front. I dislike Burn. I dislike it mechanically in 2E, and I absolutely disagreed with its implementation in 1E. I think it was overly complex, tried to solve the wrong problems, and didn't succeed at its own goals.
For those not familiar with burn, I won't dilute this post too much to 1E's implementation. Burn was the cost you paid when you used most 1E kineticist abilities. You paid modular amounts of burn to use these abilities, and each point represented taking nonlethal damage equal to your character level in a way that wouldn't heal until you rested. Burn is mentioned 53 times in the Archives of Nethys front page for the class, and is notoriously complex. I recommend reading what others have already stated about it.
Anyway, let's talk about one 2E and burn. I've seen two simple suggestions for Burn in 2E:
Burn = take nonlethal damage equal to your level
Burned X = drained X
There is a very simple mechanical problem with either of these in the context of 2E: ancestry HP. A level 1 character can take up to 22 instances of either type of Burn, while that same character at level 2 would only be able to take 17. This number will keep going down.
The second mechanical problem is that 2E is designed around easy access to out of combat healing. In the vast majority of fights, the party is assumed to be at full HP. This negates the danger of taking damage, but makes losing maximum Hit Points much more serious.
The third mechanical issue is less pressing. Nonlethal damage works completely differently. In 1E, it was tracked separately and gave you a cushion of "I'll go down, but I'll just pass out." In 2E, the nonlethal trait does nothing unless it is the last bit of damage you take before going to 0 HP.
These mechanical issues make it very easy to undershoot or overshoot a HP based cost in this system, or to have a cost gets worse as a player levels.
Anyway, I think there are some key questions that need to be asked of any proposal to add Burn back in as a core feature for the kineticist:
What are you trying to achieve?
If merely adding a non-Focus Point cost to using features, does your implementation effectively even work with 2E's healing?
Is it fun to manage your version of burn?
Do you account for ancestry Hit Points? Does your version work worse at level 2?
Is your version simple enough to understand in one reading?
Do you need a lot of class features (and page space) to make your version of Burn integrate into the class?
Does your version of burn account for people that want cool Wuxia/Avatar shenanigans and want to opt-out of burn-like systems? (Trust me, I heard from plenty of people like this in my attempt.)
To be clear, I would rather see Burn left out entirely than to see it poorly thrown in. And I mostly see it as a justification for keeping Constitution as the main stat and bumping the damage numbers/action economy across the board, which I don't feel answer any of the questions above.
For reference, here is how my systems tried to answer these questions. I'll note that my goal was to appease both camps: people who wanted burn as a power-fantasy, and people that just wanted cool elemental shenanigans and big HP pool. I'll note that my first version is an almost identical chassis to what the 2E version ended up as, so it is much more comparable.
When I made my first kineticist homebrew, these were major issues that came up in playtesting. My solution wasn't elegant. You would ramp up from unburned, to burning, to blazing, to overcharged. These states dealt damage equal to 1/2/3 + your level, and that damage couldn't be healed unless you had been unburned for 10 minutes.
This meant that a level 1 character could be overcharged for up to 5 rounds, while a level 20 character could be overcharged for 13 rounds. The conditions themselves gave varying bonuses; overcharged gain +3 damage per damage dice on your kinetic blast, a +2 status bonus to your kineticist class DC, and a +2 status bonus to your element's skills.
My subclasses were based around embracing or avoiding Burn: Cautious Disciple kineticists could easily reduced their state to unburned and gain temporary hit points while reckless disciple kineticists could overcharge themselves and skip the turn by turn ramping.
I think that the current playtests subclasses (the gate types) are more flavorful and less mechanically tricky to balance.
My second version was almost completely different from the playtest, and less well received. It was just a spellcaster that used a much simpler version of burned with no internal benefit, and that version was very close to the burned X = drained X implementation.
There are points to be made about simplicity, but this didn't play all that well and didn't have much internal logic behind it.
