When is pullback by a quasi-isomorphism a quasi-isomorphism again? Given two chain complexes $A,V$, one has a chain complex $\hom(A,V)$ by putting in degree $n$ the maps $A\to V$ of degree $n$, and defining the differential by
$$\partial(\varphi) := d_V\varphi - (-1)^\varphi\varphi d_A\ .$$
My question is the following one.

Let $f:A\to B$ be a quasi-isomorphism, and let $V$ be another chain complex. Under what conditions is the pullback
  $$f^*:\hom(B,V)\longrightarrow\hom(A,V)$$
  a quasi-isomorphism? And what about the pushforward?

It seems weird to me that I haven't been able to find a standard answer to this question around. Maybe I haven't looked in the correct places.
It might well be that the statement is always true (I haven't come up with a counterexample yet, at least). I would like an answer in terms of "common" properties for linear maps (e.g. injectivity, surjectivity, ...) if possible.
One intuitive way to try to attack the problem is via model categories. However, we have inner homs here, and not the usual hom sets, and thus I am a bit at loss about how to translate the problem in the correct way.
I also have the following easy partial result:
Def: A homotopy retraction of chain complexes from $X$ to $Y$ is a triple of maps
$$(h:X\to X,p:X\to Y,i:Y\to X)$$
such that
$$pi=1_Y,\qquad 1_X-ip=d_Xh+hd_X\ .$$
In particular, the maps $i$ and $p$ are quasi-isomorphisms.
Lemma: If $f=i$ or $f=p$ in a homotopy retraction, then $f^*$ and $f_*$ are quasi-isomorphisms (without any conditions on $A,B$ and $V$).
 A: If your chain complexes are over the additive category $\mathcal{A}$, then the degree $n$ (or $-n$, depending on your conventions) homology of $\text{hom}(A,V)$ is $\text{Hom}_{K(\mathcal{A})}(A,V[n])$, where $K(\mathcal{A})$ is the homotopy category of chain complexes, and $-[n]$ denotes shift in degree by $n$.
If $C_f$ is the mapping cone of $f$, then there is a long exact sequence
$$\dots\to\text{Hom}_{K(\mathcal{A})}(A,V[n-1])
\to\text{Hom}_{K(\mathcal{A})}(C_f,V[n])
\to\text{Hom}_{K(\mathcal{A})}(B,V[n])
\to\text{Hom}_{K(\mathcal{A})}(A,V[n])
\to\text{Hom}_{K(\mathcal{A})}(C_f,V[n+1])\to\dots$$
and so $f^*$ is a quasi-isomorphism if and only if $\text{Hom}_{K(\mathcal{A})}(C_f,V[n])=0$ for all $n$, or equivalently $\text{hom}(C_f,V)$ is acyclic.
This will always be the case, independent of $V$, if $f$ is a chain homotopy equivalence, generalizing the case of a homotopy retraction in the question.
For an example where it's not the case, let $A$ be any acyclic, but not contractible, complex, $f:A\to 0$ the zero map, and $V=A$. Then $\text{hom}(0,V)=0$, but $\text{hom}(A,V)$ has homology $\text{Hom}_{K(\mathcal{A})}(A,A)$ in degree zero, which is non-zero since the identity map $A\to A$ is a non-zero element.
