I am not sure if you are only really interested in properly stacky things, but it is perhaps worth pointing out that the result you mentioned from Hartshorne is true in significantly greater generality.

For any quasi-compact quasi-separated scheme $X$ (in fact for any spectral space $X$, or for something even slightly weaker) and any filtered system $(\mathcal{F}_\alpha)$ of sheaves of abelian groups on $X$ there is an isomorphism $$\mathrm{colim}\; H^i(X, \mathcal{F}_\alpha) \cong H^i(X, \mathrm{colim}\; \mathcal{F}_\alpha)$$ for all $i \geq 0$.

I have a feeling this is well known (as usual to those who well know it), but at least one place this is written down is in a paper of Kempf, more specifically in Theorem 8.

I am not sure if you are only really interested in properly stacky things, but it is perhaps worth pointing out that the result you mentioned from Hartshorne is true in significantly greater generality.

For any quasi-compact quasi-separated scheme $X$ (in fact for any spectral space $X$, or for something even slightly weaker) and any filtered system $(\mathcal{F}_\alpha)$ of sheaves of abelian groups on $X$ there is an isomorphism $$\mathrm{colim}\; H^i(X, \mathcal{F}_\alpha) \cong H^i(X, \mathrm{colim}\; \mathcal{F}_\alpha)$$ for all $i \geq 0$.

I have a feeling this is well known (as usual to those who well know it), but at least one place this is written down is in

• George R. Kempf, Some elementary proofs of basic theorems in the cohomology of quasi-coherent sheaves, Rocky Mountain J. Math. 10(3) (1980) pp. 637-646, doi:10.1216/RMJ-1980-10-3-637,

more specifically in Theorem 8.