The earlier replies to this question established the implausibility of drawing on the zero point energy for practical use. Matt Visser of Washington University in St. Louis adds some technical details:
The Zero Point Energy (ZPE) is an intrinsic and unavoidable part of quantum physics. The ZPE has been studied, both theoretically and experimentally, since the discovery of quantum mechanics in the 1920s and there can be no doubt that the ZPE is a real physical effect. The "vacuum energy" is a specific example of ZPE which has generated considerable doubt and confusion. In a completely empty flat universe, calculations of the vacuum energy yield infinite values of both positive and negative sign--something that obviously does not correspond to the nature of the real world.
Observation indicates that in our universe the grand total vacuum energy is extremely small and quite possibly exactly zero. Many theorists suspect that the total vacuum energy is exactly zero.
It definitely is possible to manipulate the vacuum energy. Any objects that change the vacuum energy (electrical conductors, dielectrics and gravitational fields, for instance) distort the quantum mechanical vacuum state. These changes in the vacuum energy are often easier to calculate than the total vacuum energy itself. Sometimes we can even measure these changes in the vacuum energy in laboratory experiments.
In classical physics, if you have a particle that is acted on by some conservative force, the total energy is E = (1/2) mv2 + V(x). To find the classical ground state, set the velocity to zero to minimize the kinetic energy, (1/2)m v2, and put the particle at the point where it has the lowest potential energy V(x). But this result is only a classical approximation to the real world. Because the classical ground state completely specifies both the particle's speed (zero) and position (at the minimum), it violates the famous Heisenberg Uncertainty Principle (m dv dx > hbar). Quantum physics, via the Uncertainty Principle, forces the particle to spread out both in position and velocity and so causes it to have an energy somewhat higher than the classical minimum. The ZPE is defined as this shift:
particle would have if we were to give it a small push. Quantum mechanically, it is now an undergraduate exercise to use the Heisenberg uncertainty relation (more precisely, Schroedinger's differential equation) to show that
The next step is to realize that the electromagnetic field can be thought of as an infinite collection of coupled oscillators--one at each point in space. Again, the classical ground state is the case in which the electric and magnetic fields both must be zero. Quantum effects mean that this case does not hold true; there is also a Heisenberg uncertainty principle for electric and magnetic fields (it's a little more complex). The good news is that the potential for electromagnetism is exactly quadratic and so can be solved exactly. The bad news is that there is an infinite number of modes. Formally we can write
The first and most obvious problem is that there are other quantum fields in the universe apart from electromagnetism. Electrons, for starters, plus neutrinos, quarks, gluons, W, Z, Higgs and so on. In particular, if you do the calculation for electrons you will find that what are known as Fermi statistics give rise to an extra minus sign in the calculation.