Adding minus infinity to plus infinity gives mathematicians nightmares and even makes theoretical physicists worry a little. Fortunately, nature does not worry about what the mathematicians or physicists think and does the job for us automatically. Consider the grand total vacuum energy (once we have added in all quantum fields, all particle interactions, kept everything finite by hook or by crook, and taken all the proper limits at the end of the day). This grand total vacuum energy has another name: it is called the "cosmological constant," and it is something that we can measure observationally.
In its original incarnation, the cosmological constant was something that Einstein put into General Relativity (his theory of gravity) by hand. Particle physicists have since taken over this idea and appropriated it for their own by giving it this more physical description in terms of the ZPE and the vacuum energy. Astrophysicists are now busy putting observational limits on the cosmological constant. From the cosmological point of view these limits are still pretty broad: the cosmological constant could potentially provide up to 60 percent to 80 percent of the total mass of the universe.
From a particle physics point of view, however, these limits are extremely stringent: the cosmological constant is more than 10(-123) times smaller than one would naively estimate from particle physics equations. The cosmological constant could quite plausibly be exactly zero. (Physicists are still arguing on this point.) Even if the cosmological constant is not zero it is certainly small on a particle-physics scale, small on a human-engineering scale, and too tiny to be any plausible source of energy for human needs--not that we have any good ideas on how to accomplish large-scale manipulations of the cosmological constant anyway.
Putting the more exotic fantasies of the free lunch crowd aside, is there anything more plausible that we could use the ZPE for? It turns out that small-scale manipulations of the ZPE are indeed possible. By introducing a conductor or a dielectric, one can affect the electromagnetic field and thus induce changes in the quantum mechanical vacuum, leading to changes in the ZPE. This is what underlies a peculiar physical phenomenon called the Casimir effect. In a classical world, perfectly neutral conductors do not attract one another. In a quantum world, however, the neutral conductors disturb the quantum electromagnetic vacuum and produce finite measurable changes in the energy as the conductors move around. Sometimes we can even calculate the change in energy and compare it with experiment. These effects are all undoubtedly real and uncontroversial but tiny.
More controversial is the suggestion, made by the physicist Julian Schwinger, that the ZPE in dielectrics has something to do with sonoluminescence. The jury is still out on this one and there is a lot of polite discussion going on (both among experimentalists, who are unsure of which of the competing mechanisms is the correct one, and among theorists, who still disagree on the precise size and nature of the Casimir effect in dielectrics.) Even more speculative is the suggestion that relates the Casimir effect to "starquakes" on neutron stars and to gamma ray bursts.
In summary, there is no doubt that the ZPE, vacuum energy and Casimir effect are physically real. Our ability to manipulate these quantities is limited but in some cases technologically interesting. But the free-lunch crowd has greatly exaggerated the importance of the ZPE. Notions of mining the ZPE should therefore be treated with extreme skepticism
From the way some enthusiasts talk about the zero-point energy, one might think that unlimited power is lying all around just waiting to be harnessed. Like many ideas that seem too good to be true, this one falls apart on closer examination, although the concept of the zero-point energy is quite fascinating in and of itself. John Obienin, a materials science researcher at the University of Nebraska at Omaha, explains: