I've not looked very far into zero-point energy, but, assuming it's thermodynamically valid as you say, wouldn't truly harnessing it lower the zero point of the particles involved to balance the system?

Therefore, you'd either run out of energy when all the zero-points became literal zero, find that the energy harnessed is linear while the zero-point drop is inverse-exponential (an amazing fact indeed), or find that the zero-point is able to become negative.

Don't think of it as an infinite gas tank, but rather a gas tank that's roughly the size of the universe. Theoretically you'll empty the tank eventually, but it'll take a good while.