Although wurtzite InP nanowires have recently been grown, an accurate description of the wurtzite InP phonon dispersion is still missing. We calculate the ab initio phonon dispersion of wurtzite and zinc-blende InP using density-functional perturbation theory and a real space supercell approach. Our predicted optical phonon frequencies agree well with measured Raman data from InP nanowires. We find that treating In 4d electrons as valence electrons is required to accurately describe InP lattice dynamics and dielectric constants, but including spin-orbit coupling has little effect. We also compare the sound velocities and specific heat and find that any difference in the thermal conductivity of InP polytypes should be due to differences in phonon-scattering rates.