To answer we need to clarify something about how a Toa’s Elemental Powers work.
Firstly, Toa use a special kind of energy called Elemental Energy (not sure if the term itself is canon but Greg has used it so I will too). This lets them create and manipulate their element. Toa can also absorb their own element for power, but Greg has said (I’ll find the quotes later) that absorbing one’s element is unstable, and actually requires Elemental Energy to do, so they can’t just endlessly recycle it. When Vakama Metru absorbed loads of heat energy (will find source later), he had to release that shortly after, because he couldn’t simply turn that heat back into Elemental Energy, and his body just couldn’t contain that non-Elemental Energy within his body for long. Elemental Energy just recharges itself over time. This is why the Toa Metru didn’t use their Elemental Powers in Legends of Metru Nui, since they were drained from fighting the Morbuzahk and were still recharging. It seems like fully recovering takes something in the realm of a few days.
(Sidenote, I don’t remember what his response to questions about Gali Nuva seemingly recharging her powers with a stream in Mask of Light were, but I think he has addressed that somewhere too, I’ll look that up later as well)
Regarding the Nui Stone, I would imagine that the heat it releases is just a by-product of it containing that much power, sort of like leaving a lightbulb on. So if a Toa of Fire absorbed that heat, they wouldn’t be leeching off the Nui Stone’s actual power, just a by-product that’d be a tiny fraction of the energy contained within.
Even if the Nui Stone emits enough heat for it to be worth absorbing for a Ta-Toa, doing so wouldn’t replenish their Elemental Energy reserves, it would actually make the reserves go down faster since they’re using them to do the absorption in the first place.
Also even if a Ta-Toa could regain their own lost energy in a perfectly efficient endless loop, the Nui Stone could easily have the energy from other Toa stored within it too.
Now, I can imagine that a Toa of Fire absorbing the heat off the Nui Stone might affect it in some way if they made it cold enough, but I have no idea what that would be, if anything.
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@ProfSrlojohn Toa of Psionics are an exception, in that they cannot absorb Psionics energy (it’s on BS01 (also I guess Greg thought that’d be OP)). The rest can absorb their own element though.
That does raise interesting questions about what that would look like, though. Absorbing sound would presumably just “create” silence, but what does absorbing Shadow look like? Would that brighten a room in effect? That one never really made sense to me…
Edited for Double Post- Prentice1215
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@Wolk I didn’t actually know that, thanks for pointing that out. That being said, Elemental Energy is dependant on a Toa being a Toa, which is mostly dependant on them having Toa Power. Regarding the OP, any heat they absorb wouldn’t replenish the Toa Energy either, so it still results in the answer being that Ta-Toa wouldn’t be invulnerable to the Nui Stone’s effects.