(This mostly just adds some references to Tyler's answer)

$\newcommand{\X}{\mathcal X}$
As Tyler said, if $F$ is a quasi-coherent $\cal O$-module, you get the Grothendieck spectral sequence

$$H^p(X,R^q\pi_*(F))\Rightarrow H^{p+q}(\X,F)$$

so it suffices to impose the condition that $R^q\pi_*=0$ for $q>0$ (i.e. that $\pi_*$ is exact). This is exactly the condition that $\X$ is *tame*. See Abramovich-Olsson-Vistoli's Tame stacks in positive characteristic. In particular, DM stacks in characteristic 0 are always tame.^{†}

Note that it didn't matter that $\pi$ was a coarse space map. We just needed that $\pi_*$ is exact (that "$\pi$ is *cohomologically affine*"). In particular, the isomorphism holds when $F$ is quasi-coherent and $\pi:\mathcal X\to X$ is a good moduli space.

^{†} This is asserted in the first paragraph of the paper, but I don't see how to prove it. Perhaps somebody could clarify in a comment.