We calculate the classical-mechanical vibrational line shape of a polar, harmonic diatomic molecule dissolved in a fused salt. This model system is chosen to allow exploration of the effects of Coulomb interactions on vibrational dynamics. The Coulomb interaction is scaled by an effective dielectric constant, whose magnitude is varied to alter relative contributions of long- and short-ranged forces to the line shape. Calculations are performed with an analytical theory based on the instantaneous normal mode approximation, in which the dynamics of a liquid are mapped onto the motions of a disordered harmonic network. Results are compared to calculations based on molecular dynamics simulations of a fused salt. The vibrational line shape predicted by this approach may be expressed in the form associated with the generalized Langevin equation, allowing the identification of a frequency-dependent bond friction. The contribution to this generalized friction coefficient from long-ranged electrostatic interactions is analyzed.