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?



Faithfulness of derived functor.
This will usually not be the case. For example, consider a "typical forgetful functor" for example


$$A-Mod \rightarrow k-vect$$


from the category of representations of a k-algebra $A$ to vectorspaces. 
It is exact faithful, but its derived functor will only be faithful if the algebra is semi-simple, since there are no Ext between vector spaces. For a concrete example take $A=k[x]$.


More philosophically, I think of an exact faithful functor as a functor which forgets some additional structure. The question whether $DF$ is faithful is the question whether there are no new extensions coming from the additional structure. This will rarely be the case.


