Sheaf cohomology commutes with colimits of sheaves

I think the canonical reference (true for any coherent topos, so for every qcqs scheme, as Remy mentions) is SGA 4 II, Expose VI, Corollaire 5.2.

Let $E'$ be a coherent topos. For every integer $q$, the functor $H^q(E', -)$ commutes with filtered inductive limits of abelian sheaves.

Hartshorne Chapter 3 Proposition 2.9

can be apply to this case