With respect to the Casimir effect, why can't the wavelengths of the virtual particles between two plates just "pass through" the plates themselves? I've read over the years that the suppression of many of the possible wavelengths between the two plates in a Casimir experiment is what causes the phenomenon (top comment on this Askscience thread is one example of this explanation). What exactly is the mechanism by which these modes aren't allowed? Are the atoms in the material actually inhibiting the creation of certain virtual particles? I would figure that virtual particles would still be doing their thing even within matter, not just around it in empty space. Is this just a useful heuristic? And if not, could someone please explain the exact mechanism that suppresses any and all modes from being allowed within the space of the two plates?
Any help understanding this is appreciated.
 A: Be aware that the term "virtual particle," although very common, is ambiguous and potentially misleading.  What's relevant here are the electromagnetic field modes (ordinary functions, which are coefficients of the quantum creation and annihilation operators), which are one thing people mean by "virtual particles."  
These modes are indeed affected by the matter they propagate through.  As a first approximation, you can think of these modes as responding to the matter just like a classical electromagnetic field responds to a dielectric.  (Really, we should look at a full quantum theory of the charges in the matter and the electromagnetic field.)  In the limit of perfectly reflecting walls, then, these modes satisfy the usual perfect-reflection boundary conditions.  Even without this limit, the density of the modes, as a function of wave-vector and polarization, is different in the Casimir case from what it would be in an unbounded vacuum.  In turn, this means the renormalized vacuum energy density is different, and this is what gives the famous explanation for the Casimir effect.
A: This is in addition to the chosen answer, to clarify the term "virtual" as used in particle physics:
Virtual particles carry the particle name but not the mass. They are off mass shell and are a mathematical tool in calculating crossections, lifetimes etc at the microscopic level where quantum mechanics reigns. The answer of fields filling up space with creation and annihilation operators is mathematically correct but imo confusing. Physics is about concrete numbers and the connection of fields in space to numbers is not intuitive unless one has worked into calculating with Feynman diagrams 
Have a look at this Feynman diagram 

The particle between the two vertices carries the title "gamma" because it has all the quantum numbers of the gamma except its mass, which can vary according to the boundaries of integration . The diagram is a recipe for getting the integral which will give the crossection for the process, after integration. Integration means there are boundaries defined by the problem under consideration. 
Were the Casimir effect written down as a convolution of the individual Feynman diagrams that contribute to it the boundary conditions would be very important to all the virtual particles, including the ones that represent the vacuum. 

I would figure that virtual particles would still be doing their thing even within matter, not just around it in empty space 

They always exist in the calculations to compare with real life, but the boundary conditions of the specific problem give different wavefunctions to close the integral which will give the real numbers for any interaction. Thus "virtual" depends on the problem under consideration.
A: 
With respect to the Casimir effect, why can't the wavelengths of the virtual particles between two plates just “pass through” the plates themselves?

Did you mean vacuum fluctuations? They aren't the same thing as virtual particles. Virtual particles are "field quanta". It's like you divide up the electron's electromagnetic field into abstract chunks and say each one is a virtual particle. Then when the electron and the proton attract each other, the resultant hydrogen atom doesn't have much of a field because the electron and the proton have "exchanged field". Vacuum fluctuations are different, they're the electromagnetic equivalent of the little ripplets on the surface of the sea. They're real, not virtual. And they can't pass through the plates because the plates are made out of metal. Check out electromagnetic shielding. 

I've read over the years that the suppression of many of the possible wavelengths between the two plates in a Casimir experiment is what causes the phenomenon (top comment on this Askscience thread is one example of this explanation). What exactly is the mechanism by which these modes aren't allowed? 

It's just the usual standing wave thing, where you can only have a whole number of wavelengths in the cavity. See this little Scientific American article. 

Are the atoms in the material actually inhibiting the creation of certain virtual particles? 

No, it's more like the metal acts like a mirror, see these lecture notes by Hans Martin Schmid.  

I would figure that virtual particles would still be doing their thing even within matter, not just around it in empty space. Is this just a useful heuristic? 

Probably not for the Casimir effect.  

And if not, could someone please explain the exact mechanism that suppresses any and all modes from being allowed within the space of the two plates?

It's nothing special, think guitar and see hyperphysics, or read the Wikipedia article: "This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. The most common cause of standing waves is the phenomenon of resonance, in which standing waves occur inside a resonator due to interference between waves reflected back and forth at the resonator's resonant frequency..." 
