A theory is presented for the dispersion relations of the nonlinear phonon-polaritons arising when phonons are coupled to the electromagnetic waves in multilayered structures of nonlinear materials. The calculations are applied to a multilayered structure consisting of a thin film surrounded by semi-infinite bounding media where each layer may have a frequency dependent dielectric function and Kerr-type nonlinearity. At least one of the media is an ionic crystal supporting optical phonon modes. The resulting analytic and numerical solutions for the dispersion relations of phonon-polaritons with s-polarization are considered for several cases. We find that the presence of nonlinearities leads to multiple branches in the dispersion relation. The results are plotted as frequency versus wave vector and frequency versus nonlinearity for different phonon-polariton modes. The parameters that modify the modes correspond to the in-plane wave vector, the thickness of the film, the phonon frequencies and the nonlinearity of each layer.