Let $F:{\cal A}\to {\cal B}$ be an additive, exact and faithful functor between abelian categories. Then on the level of complexes, $F$ maps quasi-isomorphisms to quasi-isomorphisms and thus induces a functor on the derived categories $DF:D({\cal A})\to D({\cal B})$.

Is it true that $DF$ is faithful? Likewise for the usual categories of bounded complexes. If not, are there extra conditions that guarantee faithfulness?