I think that it only happens in trivial cases. Let me assume that $\mathcal{C}$ has binary products and coproducts. The functor $\mathrm{Hom}_{\mathcal{C}}$ preserves products if and only if for all $A,A',B,B' \in \mathcal{C}$ the canonical map
$$\mathrm{Hom}(A+A',B \times B') \longrightarrow \mathrm{Hom}(A,B) \times \mathrm{Hom}(A',B')$$
is an isomorphism. However, the left hand side decomposes as
$$\mathrm{Hom}(A,B) \times \mathrm{Hom}(A,B') \times \mathrm{Hom}(A',B) \times \mathrm{Hom}(A',B')$$
and the canonical map is simply the projection. In particular, the projection
$\mathrm{Hom}(A,B)^4 \to \mathrm{Hom}(A,B)^2$ is injective, which shows that $\mathrm{Hom}(A,B)$ has at most one element. Thus, $\mathcal{C}$ is a preorder. The surjectivity of
$$\mathrm{Hom}(A+A',A \times A') \longrightarrow \mathrm{Hom}(A,A) \times \mathrm{Hom}(A',A')$$
shows that $A+A' \leq A \times A'$, hence $A' \leq A$ for all $A,A'$. Thus, $\mathcal{C}$ is a trivial (codiscrete) preorder.