Given a discrete torsion-free group $G$, we define the cohomological dimension ($cd$) of $G$ to be the minimum $N$ such that $H^{n}(G,M)=0$ for all $\mathbb{Z}G$-modules $M$ and all $n>N$. This is extended to groups with torsion as virtual cohomological dimension ($vcd$): the virtual cohomological dimension of $G$ is the cohomological dimension of a torsion-free subgroup of finite index (that this is well defined is not immediately obvious, but true). While it is interesting that this is a well defined invariant, does this contain any more information?
For example, if $G$ has $vcd(G)=1$, does this tell us that $H^{2}(G,M)$ vanishes for all $\mathbb{Q}G$-modules, or something similar? In the case of $G$ a cocompact lattice in $SL_2(\mathbb{R})$, then we can establish this by computation of the cohomology of the associated orbifold quotient (see for example Harder's Cohomology of Arithmetic Groups) but this approach is not available in general. Can we say anything meaningful about group cohomology knowing virtual cohomological dimension in general?
I am turning my comment into an answer.
Let $G$ have virtual cohomological dimension $N$. Let $M$ be a $\mathbb{Z}[G]$-module and $n > N$. Let $H \leq G$ be a torsion-free subgroup of finite index and cohomological dimension $n$.
Then the sequence $H^n(G,M) \rightarrow H^n(H,M) \rightarrow H^n(G,M)$ (restriction then corestriction) is formally $[G:H]$.
But our assumptions imply that $H^n(H,M)=0$: therefore $[G:H] \cdot H^n(G,M)=0$. In particular, if $M$ is a $\mathbb{Q}$-vector space, then $H^n(G,M)=0$.