Multiferroic rare-earth manganites are theoretically studied by focusing on the coupling to the lattice degrees of freedom. We demonstrate analytically that the phonon excitations in the multiferroic phase are strongly affected by the magnetoelectric coupling, the spin-phonon interaction and the anharmonic phonon-phonon interaction. Based on a microscopic model, the temperature dependence of the phonon dispersion relation is analyzed. It offers an anomaly at both the ferroelectric and the magnetic transition indicating the mutual coupling between multiferroic orders and lattice distortions. Depending on the sign of the spin-phonon coupling the phonon modes become softer or harder in accordance with experimental observations. We show that the phonon spectrum can be also controlled by an external magnetic field. The phonon energy is enhanced by increasing that field. The applied Green's function technique allows the calculation of the macroscopic magnetization depending on both the phonon-phonon and the spin-phonon couplings.