Let $F: \mathcal{A} \rightarrow \mathcal{B}$ be an additive functor between abelian categories (with enough injectives and projectives) and $K^\cdot, L^\cdot$ objects of $\textrm{Ch}(\mathcal{A})$. Suppose $F$ is left (resp. right) exact. Then $F$ preserves quasi-isomorphisms between complexes $f: K^\cdot \rightarrow L^\cdot$ of objects acyclic for $F$. Since $\textrm{Cyl}(f)$ is quasi-isomorphic to $L^\cdot$ and the objects of $\textrm{Cone}(f)$ are a direct sum of those in $L^\cdot, K^\cdot$, this follows by applying $F$ to the short exact sequence
$0 \rightarrow K^\cdot \rightarrow \textrm{Cyl}(f) \rightarrow \textrm{Cone}(f) \rightarrow 0$ (p. 155, Gelfand-Manin).
Is the same true in more generality? I know that only exact functors preserve arbitrary quasi-isomorphisms, but do left (resp. right) exact additive functors preserve lots of other quasi-isomorphisms? (Feel free to impose boundedness conditions on the relevant complexes $K^\cdot, L^\cdot$ and any reasonable conditions on $\mathcal{A}, \mathcal{B}$).
Your question is unclear at some points, e.g. what are your complexes made of? what's the source/target of $F$? ... –  Fernando Muro Oct 2 '12 at 22:23