The phonon dispersion was measured at room temperature along (0,0,L) in the tetragonal phase of LaFeAsO using inelastic x-ray scattering. Spin-polarized first-principles calculations imposing various types of antiferromagnetic order are in better agreement with the experimental results than nonmagnetic calculations, although the measurements were made well above the magnetic ordering temperature, T_N. Splitting observed between two A_1g phonon modes at 22 and 26 meV is only reproduced in spin-polarized calculations. Magnetostructural effects similar to those observed in the AFe₂As₂ (A=Ca,Sr,Ba,Eu) materials are present in LaFeAsO. The inclusion of Fe spin is necessary to find reasonable agreement of the calculations with the measured spectrum well above T_N. On-site Fe and As force constants show significant softening compared to nonmagnetic calculations, however an investigation of the real-space force constants associates the magnetoelastic coupling with a complex renormalization instead of softening of a specific pairwise force.