I know of one very limited case, which will almost certainly not satisfy you, where the "virtual cohomology" mentioned by Angelo exists, with a version of Poincare duality.

Presumably the generality one would like to address this question is for a moduli space (or stack) $M$ carrying a perfect obstruction theory (let me ignore the stackyness everywhere below). Let me restrict right away to virtual dimension 0, and in fact further to the case when the obstruction theory is symmetric (in the sense of Behrend, Behrend-Fantechi). Note that this already excludes Kontsevich moduli spaces $\bar M_{g,n}(X,\beta)$, even for $X$ a Calabi--Yau threefold, but we are still OK with DT- or other sheaf-theoretic moduli spaces.

In this case, we of course have Behrend's result that locally $M$ is cut out in a smooth ambient space $N$ by the zeros of an almost closed one-form. Now specialize further, and assume that in fact $M$ is cut out by the zeros of a closed one-form. Finally, specialize to the case when in fact globally $M=Z(df)\subset N$; here $N$ is a smooth variety, $f\colon N\to {\mathbb C}$ is a smooth function, and $M$ is cut out by the zeros of the exact one-form $df$. Note that there are still reasonably interesting examples, such as the Hilbert scheme of points on ${\mathbb C}^3$ or related toric or quivery examples.

In this case, it seems that the correct "virtual cohomology" to consider is $H^*(M,\phi_f)$, the cohomology of $M$ with coefficients in the perverse $\mathbb Q$-sheaf $\phi_f\in{\rm Perv}_{\mathbb Q}(M)$ of vanishing cycles of $f$ (appropriately shifted). This is called critical cohomology by Kontsevich-Soibelman. For example, this cohomology has the correct Euler characteristic, namely the integral of the Behrend function on $M$. It also carries a mixed Hodge structure, and the Euler characteristic of the weight filtration (which will differ from the degree filtration because the Hodge structure is not pure) gives what seems to be the right "quantization" (or "refinement") of the numerical invariant.

Now the point is that by standard theory, $D_M\phi_f\cong\phi_f$, where $D_M$ is the Verdier duality functor on $M$. This will give you a kind of Poincare duality
$$H^n(M,\phi_f)\cong H^{-n}_c(M,\phi_f)$$
albeit between ordinary and compact-support cohomology, since $M$ is of course almost always noncompact. But at least (using the appropriate shifts) it is elegantly symmetric around zero, which is what we would expect in the virtual dimension 0 case.

I don't really know how well this generalizes; almost nothing is known (to me) about gluing, functoriality, etc.