  We consider rare semileptonic decays of a heavy $D$-meson into a light vector
meson in the framework of QCD factorization. In contrast to the corresponding
$B$-meson decays, the naive factorization hypothesis does not even serve as a
first approximation. Rather, the decay amplitudes appear to be dominated by
non-factorizable dynamics, e.g. through annihilation topologies, which are
particularly sensitive to long-distance hadronic contributions. We therefore
pay particular attention to intermediate vector-meson resonances appearing in
quark-loop and annihilation topologies. Compared to the analogous $B$-meson
decays, we identify a number of effects that result in very large theoretical
uncertainties for differential decay rates. Some of these effects are found to
cancel in the ratio of partially integrated decay rates for transversely and
longitudinally polarized $\rho$ mesons. On the phenomenological side this
implies a very limited potential to constrain physics beyond the Standard Model
by means of these decays.
