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..."