Does energy conservation apply to Casimir effect? If you cancel out some quantum field modes using two 'Casimir' plates you decrease the average energy density in the region and gain potential energy in Casimir force approximately proportional to the plate areas and inversely proportional to plate separation to the fourth power? Is it enough to make up for the supposed loss of energy in the form of vibrational energy of quantum fields?
 A:  Short answer: yes 
Yes, energy conservation does apply to the Casimir effect. The Casimir effect is present in ordinary QFT in flat spacetime, in which the Hamiltonian (the total energy operator) is independent of time, so energy is conserved in every situation, including situations involving the Casimir effect.
 Using what we're able to calculate to draw inferences about what we wish we could calculate 
Energy conservation says that the total energy of the system does not change with time. Calculations of the Casimir effect typically use a short-cut in which the material plates (which would be made of some complicated arrangement of electrons and nuclei) are replaced with idealized boundary conditions on space itself. In that short-cut, the "force" between the boundaries of space is defined in terms of $dE/dx$, where $E$ is the energy of the ground state (with the given boundary conditions) and $x$ is the distance between the boundaries. Because of the nature of this short-cut, the concept of energy conservation doesn't strictly apply, because the whole situation is static (not time-dependent). That's a trivial form of energy conservation, not what the OP is really asking about. 
To see the form of energy conservation that the OP is asking about in an explicit calculation of the Casimir effect, we would need to consider material plates constructed from dynamic ingredients (quantum fields, like electron and nucleon fields), and the calculation suddenly becomes prohibitively difficult. Even constructing a state that describes such material plates is prohibitively difficult, not to mention calculating its time-dependence.
However, if we assume that a theory like QED + QCD is sufficient for constructing such a state, and if we assume that the boundary-condition short-cut calculation is a fair proxy for the Casimir force between actual material plates, then we can immediately conclude that the Casimir effect conserves energy. Energy is manifestly conserved in QED + QCD, where the Hamiltonian is explicitly known and explicitly time-independent. Recall that the Hamiltonian is both the total energy operator and the generator of time-evolution.
 No magic 
By the way, QED + QCD can be formulated rigorously in continuous time (so that energy is still exactly conserved) after replacing space with a discrete lattice. So, in principle, the aforementioned calculation could be done completely explicitly — except that it's still prohibitively difficult. However, the boundary-condition short-cut still works on a lattice (using a discrete-derivative version of $dE/dx$), and the short-cut still leads to the conclusion that there is an attractive force between the plates (with the correct distance-dependence), so the preceding argument still holds in this perfectly well-defined formulation. 
I thought this might be worth mentioning because sometimes the Casimir effect is described using magic-sounding language about renormalization and infinities, but in the lattice formulation, everything is perfectly finite so that no magic is involved.

 Appendix: Relationship to the van der Waals force 
As emphasized in comments by safesphere, the Casimir effect is related to the van der Waals force (or is a manifestation of the van der Waals force, depending on which subculture's language we use). Since this may help put the energy-conservation question into a clearer perspective, I've incorporated the content from my comments into this appendix:
Casimir effect typically refers to an attractive interaction between closely-spaced plates. van der Waals force typically refers to an attractive interaction between neutral molecules. Whatever names we use to describe them, these are electromagnetic interactions. Words like "vacuum energy" are often used for the Casimir effect, but this over-emphasizes what I called the "short-cut" above. Describing it as an extension/special case of the van der Waals force is better. The paper https://arxiv.org/abs/hep-th/0503158 says it like this: 

The Casimir force (per unit area) between parallel plates... the standard result [which I called the short-cut], which appears to be independent of [the fine structure constant] $\alpha$, corresponds to the $\alpha\to\infty$ limit. ... The Casimir force is simply the (relativistic, retarded) van der Waals force between the metal plates.

Here's a related Physics Se post that emphasizes the same point:
Van der Waals and Casimir forces
