Here is a simple example that shows that in order to obtain a derived functor Rf_* on unbounded complexes with quasi-coherent cohomology sheaves we need some additional hypothesis beyond just requiring f to be quasi-compact and quasi-separated.

Let k be a field of characteristic p > 0. Let G = Z/pZ be the cyclic group of order p. Set S = Spec(k[x]) and let X = [S/G] be the stacky quotient where G acts trivially on S. Consider the morphism f : X —> S. Then Rf_*O_X is a complex with cohomology sheaves isomorphic to O_S for all p >= 0. In fact Rf_*O_X is quasi-isomorphic to ⊕ O_S[-n] where n runs over nonnegative integers.

Now consider the complex K = ⊕ O_X[m] where m runs over the nonnegative integers. This is an object of D_{QCoh}(X) but it isn’t bounded below. So we have to pay attention if we want to compute Rf_*K. Namely, in D(O_X) the complex K is also K = ∏ O_X[m]. Since cohomology commutes with products, we see that

Rf_*K = ∏ Rf_*O_X[m] = ∏ (⊕ O_S[m – n]).

In degree 0 we get an infinite product of copies of O_S which isn’t quasi-coherent.

Conclusion: Rf_* does not map D_{QCoh}(X) into D_{QCoh}(S).

Of course if f is a quasi-compact and quasi-separated morphism between algebraic spaces, then this kind of thing doesn’t happen.