The discovery of an enhanced superconducting transition temperature Tc in monolayers of FeSe grown on several oxide substrates has opened a different route to high-Tc superconductivity through interface engineering. One proposal for the origin of the observed enhancement is an electron-phonon (e-ph) interaction across the interface that is 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 line shape, 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.