For starters, let me say that although the Casimir effect is standard textbook stuff, the only QFT textbook I have in reach is Weinberg and he doesn't discuss it. So the only source I currently have on the subject is Wikipedia. Nevertheless I suspect this question is appropriate since I don't remember it being addressed in other textbooks
Naively, computation of the Casimir pressure leads to infinite sums and therefore requires regularization. Several regulators can be used that yield the same answer: zeta-function, heat kernel, Gaussian, probably other too. The question is:
What is the mathematical reason all regulators yield the same answer?
In physical terms it means the effect is insensitive to the detailed physics of the UV cutoff, which in realistic situation is related to the properties of the conductors used. The Wikipedia mentions that for some more complicated geometries the effect is sensitive to the cutoff, so why for the classic parallel planes example it isn't?
EDIT: Aaron provided a wonderful Terry Tao ref relevant to this issue. From this text is clear that the divergent sum for vacuum energy can be decomposed into a finite and an infinite part, and that the finite part doesn't depend on the choice of regulator. However, the infinite part does depend on the choice of regulator (see eq 15 in Tao's text). Now, we have another parameter in the problem: the separation between the conductor planes L. What we need to show is that the infinite part doesn't depend on L. This still seams like a miracle since it should happen for all regulators. Moreover, unless I'm confused it doesn't work for the toy example of a massless scalar in 2D. For this example, all terms in the vacuum energy sum are proportional to 1/L hence the infinite part of the sum asymptotics is also proportional to 1/L. So we have a "miracle" that happens only for specific geometries and dimensions