I'd be very happy if the question When do Kan extensions preserve limits/colimits? has been fully answered. But it seems not.

I have a more specific question though. Let $C$ be a site (essentially small to be safe) equipped with some subcanonical Grothendieck topology, $PSh$ be its category of presheaves. Then it's well-known that for any functor $C \to B$ with $B$ cocomplete, the left Kan extension $PSh \to B$ along the Yoneda embedding $C \to PSh$ necessarily preserves colimit. Is it true if I replace $PSh$ by the category of sheaves $Sh$? In other words, does the left Kan extension along $C \xrightarrow{Yoneda} PSh \xrightarrow{sheafification} Sh$ preserve colimit? If in general not, is there any sufficient condition we can impose on the functor $C \to B$?