Can one obtain free energy from the vacuum? It is known that from the vacuum of a quantum field theory, virtual particle pairs are created and destroyed; is it possible to capture these particles thus obtaining free energy from the vacuum? 
 A: No. 
Just like in Chemistry and Thermodynamics, we never get anything for free.
On a mechanistic level, it's important to recognize that zero-point (vacuum) energy represents the lowest energy state waveform. I remember thinking that because the EM fields are everywhere and quantized, that there was some sort of magic taking place. Realistically, zero-point energy is more like a spring between objects (very good analogy Slereah) than something unknown. It is a mathematical consequence of very well understood relations. 
The second matter of virtual particles requires a little more thought. When we expand (any) Schroedinger equation into its particle form (using the number operator (two opposing ladder operators) which changes the differential equation to a sum of a series of integrals), we get an infinite number of these terms (the virtual particles). They are a distraction at this level.
Finally, Nuclear_Wizard explained the Casimir effect brilliantly, and I won't add to that except to say that this effect represents the primary experiment validating the theory. To my knowledge and at this time, the other experiments have had a great deal of background noise to wade through (they're looking for the LOWEST energy level) and have not given nearly the experimental validation that the two plates experiment has.
One critical point that is fairly unrelated to your question (but somehow the accepted "right" answer), there is no time operator. The derivation of that relationship depends on somewhat specific circumstances relating to the decay of wavestates. If we look at a graph of the energies of incoming particles, there will be some sort of distribution - the particles don't all have the exact same energy level. If we repeat the experiment and instead look at the time of arrival (which gets us to the lifetime of the wavestate by subtraction) of particles (different setup, identically prepared source), I will also get a distribution. The Heisenberg equation (in that form) relates the uncertainty (width) of those distributions where the location of the peak represents the quantity of those distributions. There is no borrowing energy but only for a short time - the Heisenberg inequality represents a fundamental property of the Fourier Transform and NOT a vacuum energy lender.
A: The answer kinda is "You can, but why would you".
It is indeed possible to extract energy from the vacuum. It has been studied, both theoretically and experimentally, using a variety of metal plates and other Casimiresque gizmos. 
The problem is just that it basically acts like a spring. To put the Casimir effect in action, you must first approach together the two metal plates, working against the pressure of the Casimir effect. You then let the plates go, giving back said energy (minus whatever loss in the process). 
There's a number of other methods you can use for equivalent effects, such as oscillating metal plates to emit a (very weak) radiation from the Casimir effect. But none of them break energy conservation in the end. 
Although I am sure that such devices will be plenty useful for various nanotechnology things.
Edit : Oh, by the way : 
1) While the quantum vacuum is a nice explanation for it, it is (probably) not true. You can derive Casimir effects just from higher order QED effects (you can't really have a Casimir effect with metal plates because they are not perfect conductors. Though you do have the topological Casimir effect in compact spaces).
2) For some sources on various experimental gadgets using the Casimir effects, you can try "Frontiers in Propulsion Physics". It is quite a neat book that contains a lot of experimental results on such crazy topics (usually negative unfortunately!), and it contains quite a lot of references on applications of the Casimir effect.
A: The energy is borrowed from the Heisenberg Uncertainty Principle to create virtual particles and has to be paid back in a very short time.
$\Delta{t} \geq \frac{\hbar}{2\Delta{E}}$
This is why virtual particles live for very short times (i.e pop in and out of existence). We cannot manipulate this energy.
A: Whether you can extract energy from this or not (and I strongly suspect not) the Casimir effect is a consequence of vacuum fluctuations.
Essentially when two metallic plates are very close to each other, the wavelengths of virtual particles that can be created between the plates is restricted and hence there are fewer particles between the plates and outside, where no restriction occurs. This creates a pressure on the plates and pushes them together. Not a source of free energy, but an interesting (and experimentally verified) result of vacuum fluctuations.
Another well-known effect due to virtual particles is Hawking radiation. This says that when virtual particles created across the event horizon of a black hole, one can escape and the other fall in, essentially turning a virtual particle into a real particle, since it's antiparticle is inside the black hole. This is not free energy however, as the energy required comes from the mass of the black hole, causing it to (very, very) slowly evaporate over time.
So in short, no, we can't get free energy from vacuum fluctuations, but that's not to say it it doesn't have some very interesting effects.
A: No...but it would certainly be neat if true. Vacuum energy is a measure of "tension" in space time.  Over time, as the universe expands, tension diminishes.  Interestingly, the more tension, the faster light travels.  Early in the universe, after the big bang, tension was very high and the universe expanded faster than what we would consider the speed of light. In any case, tension is an expression of potential against a rest state, defined by the rest state of the initial universe.  Trying to extract energy from this potential would be analogous to trying to extract energy by connecting a wire to one pole of a battery.  Unless you can access the initial rest state of the universe, there is no way to extract energy from the vacuum.
A: Yes, but no, as any energy "gained" would not only be undetectable, but would never last. For every virtual particle, there is a virtual antiparticle. Even if you did manage to "extract" energy through somehow -impossibly- harvesting virtual particles, the antiparticles would just neutralize it. Even in a hypothetical -and very impossible- scenario where one could somehow "contain" the antiparticles, you wouldn't be able to draw any energy from it. There are several reasons for this.
1. Matter and energy can neither be created nor destroyed, only conserved, and converted into another form (e=mc². The only reason virtual particles don't violate this law, is because virtual particles and antiparticles always eradicate each other soon after forming via quantum tunnelling.
2. Virtual particles remain virtual unless their virtual antiparticle counterpart meets certain conditions: It must be inaccessible, it must not be in the same worldline, and/or must not be able to be observed. This means that in order to make a virtual particle "real", so one could harvest it, you would need to get rid of its antiparticle; which can only be done through interaction and annihilation with another particle, or through crossing an event horizon. 
3. Even if one could contain a virtual antiparticle long enough to harvest its virtual particle counterpart's energy, there would be no net gain of energy, as soon as the antiparticle would become real, it would "destroy" any matter and energy equivalent to the amount harvested from its positive energy counterpart.
It would be not only unconventional, and redundant, but would also be nothing more than a means of "destroying an object in its entirety just to harvest a small amount of equivalent energy". It wouldn't be free energy, merely a means of "quantum tunnelling shenanigins".
