A left (resp. right) exact additive functor $F$ 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 apriori preserve lots of other quasi-isomorphisms?