What happens when a the volume of a perfect vacuum (0 psi) is increased? I'm an engineer working on a design in which a "plunger" will be pulled out of a sealed vessel, where the starting volume of the vessel is essentially zero and I need to know what will happen to the surrounding structure. If I'm asking this in the wrong place then please feel free to point me in the right direction.
Say you have a hand pump with an outlet that allows you to fully depress the plunger until the volume inside is zero. The outlet is then closed and the plunger is pulled, increasing the volume inside of the pump. At some point a "perfect" vacuum will form inside of the pump and from what I understand the pressure inside the container can never drop below the inverse of pressure outside the container (1 atm). If this is true, where does the energy that is applied to the system as you continue to pull the plunger go? If the pressure can no longer decrease inside the pump then shouldn't the plunger require zero force to pull? 
 A: First off, I'd say what you would create is a "reasonable" vacuum, not a perfect vacuum.  You're creating something on par with this box moving device.
By contrast consider a vacuum like this:

A vacuum like that is not considered a perfect vacuum, but it does pull an "ultra high vacuum," which is .00000000000098 atmospheres, and involves rather exotic techniques that are quite a lot more disciplined than a simple plunger.
Now for your purposes, you can probably just assume "zero pressure" inside your vessel, and look at what forces are applied by the 1 atmosphere of pressure pushing in on it from the outside.  Practically speaking, unless the term "turbomolecular pump" or "sorption pump" mean something to you, you're probably going to treat any atmosphere left as a negligable rounding error.
Now if you're really pushing for perfect vacuums, there's a rule in the trade: everything outgasses.  Your "perfect" vacuum will get gasious atoms inside the vessel, no matter what you do.  As you push into the ultra-high vacuum world, you get fun effects to deal with, such as hydrogen atoms diffusing through your stainless steel pipes.  You can mitigate this quite a bit, but there will always be some atmosphere.
Mind you not much atmosphere.  In the ultra-high vacuum world, molecule of gas may fly 40km back and forth within your vessel before colliding with another molecule.  But there is an atmosphere.
In your case, when pulling the plunger, remember that it's not so much that the vacuum on the inside is "pulling" your plunger in, forcing you to do work.  It's more that the atmospheric pressure on the outside is "pushing" your plunger in, and there's no pressure (very little pressure) on the inside helping keep the plunger out.  You're fighting against our atmosphere.
And in theory, if you really wanted to get down to it, you'd find that if you pulled the plunger out, creating a larger vacuum, the outer edges of our atmosphere would go up slightly, and you'd find that the work you did was effectively the work required to lift that atmosphere up.
A: Suppose the plunger is 1 square inch in diameter. The work you do to pull it out is 14.7 pounds (force) times the distance you pull it out. Where's the energy? It's potential energy you got by lifting a 1 square inch column of air that distance. Let go of it, and that column of air will fall back, converting potential to kinetic energy. Then when the plunger hits the end the kinetic energy will get converted to heat.
The vacuum will not be perfect, but that won't make any practical difference. If there are 3 molecules of nitrogen in there, that's practically none.
A: First, 

from what I understand the pressure inside the container can never drop below the inverse of pressure outside the container (1 atm).

I don't know what you're referring to, but I don't believe that's the case. The pressure will continue to drop inside the vessel as you pull, but there's a limiting factor. No seal is perfect, and materials outgas as well, so gas will begin entering the vessel. But once you're at low pressure,

If the pressure can no longer decrease inside the pump then shouldn't the plunger require zero force to pull?

No. Atmospheric pressure is pushing against the vessel in all directions, so you have to worry about implosion. Even though the pressure might not decrease further in the vessel, you have to fight 14.7 psi of atmosphere in order to pull the plunger further. I.e. the increase in difficulty with which you have to pull may level off, but it will still be difficult.
A: 
I need to know what will happen to the surrounding structure.

Draw a diagram of the equipment with the pressures and forces involved.  Can the structure withstand a ~1atm pressure difference across the entire face of the container?

from what I understand the pressure inside the container can never drop below the inverse of pressure outside the container

The pressure inside can never drop below 0 pressure (absolute) or below -1 atm (gauge), assuming you're operating it at 1atm.

If this is true, where does the energy that is applied to the system as you continue to pull the plunger go?

The force you apply to the plunger moves the system into a higher potential energy configuration.  It's similar to extending a spring.  If you were to release the handle, the energy would be converted back to kinetic energy as the plunger accelerates.  

If the pressure can no longer decrease inside the pump then shouldn't the plunger require zero force to pull?

The interior pressure can not decrease below zero.  But the work you have to do is based on overcoming the net force on the plunger.  That depends on the pressure difference between the sides.  Unlike a spring that increases the resistance forces as it is extended, the plunger will have close to a constant force (ignoring minor leaks) as it is pulled.  
