Phonon polaritons localized in polar nanoresonators and superlattices are being actively investigated as promising platforms for midinfrared nanophotonics. Here we show that the nonlocal nature of the phonon response can strongly modify their nanoscale physics. Using a nonlocal dielectric approach, we study dielectric nanospheres and thin dielectric films taking into account optical phonons dispersion. We discover a rich nonlocal phenomenology, qualitatively different from the one of plasmonic systems. Our theory allows us to explain the recently reported discrepancy between theory and experiments in atomic-scale superlattices, and it provides a practical tool for the design of phonon-polariton nanodevices.