  The discovery of an enhanced superconducting transition temperature $T_c$ in
monolayers of FeSe grown on several oxide substrates has opened a new route to
high-$T_c$ superconductivity through interface engineering. One proposal for
the origin of the observed enhancement is an electron-phonon (e-ph) interaction
across the interface that peaked at small momentum transfers. In this paper, we
examine the implications of such a coupling on the phononic properties of the
system. We show that a strong forward scattering leads to a sizable broadening
of phonon lineshape, which may result in charge instabilities at
long-wavelengths. However, we further find that the inclusion of Coulombic
screening significantly reduces the phonon broadening. Our results show that
one might not expect anomalously broad phonon linewidths in the FeSe interface
systems, despite the fact that the e-ph interaction has a strong peak in the
forward scattering (small $q$) direction.
