Let k be a field and your DM-smack be [Spec(k)//G] for a trivial action of a group G. A sheaf on this stack is roughly a sheaf with a group action, and cohomology is group cohomology. If you consider a sheaf where multiplication by |G| is invertible, then group cohomology vanishes in high degrees, but otherwise there is a lot of information not coming from the cohomology of Spec(k).
EDIT: Under amenable circumstances when the left adjoint $\pi^*$ is exact, you can get a sufficient condition for your isomorphism to hold. (I'm not sure what kind of topology you want to work with, or whether you're working with simply sheaves of abelian groups or sheaves of $\mathcal{O}$-modules or quasicoherent modules or...) In this case, $\pi_*$ preserves injectives and so there's a Grothendieck spectral sequence $$ H^p(X, {\mathbb R}^q \pi_* F) \Rightarrow H^{p+q}(\mathcal{X},F) $$ and so a sufficient condition is for the higher direct image functors $\mathbb{R}^q \pi_* F$ to vanish for q > 0. These are the sheaves associated to the presheaves $U \mapsto H^q(\pi^{-1} U, F)$, and often their stalks are related to the cohomology of the fibers of $\pi$ - which are, in this case of a DM-stack, pretty much the group cohomology of the stabilizers.
