This is a simple misunderstanding about what the heat death really is. Because energy can't be destroyed, it can only be changed. It's the Law of Conservation of Energy.The heat death of the universe is when the stars run out of usable energy, so they can't keep on practicing fusion in their cores.
Also, in these rules, you can only die when it is your time to die. If you choose infinity, you can't die from a lack of energy, you can only die (or be in a coma or anything similar to death) when it is your infinityith birthday. That's the point here. After the heat death, you may starve for eternity, and go on forever thirsty, you still wouldn't shut down to that extent.
It's not a scientifically accurate thought experiment, but that's not the point.