The influence of electron-phonon interaction on the transmission phase shift of an electron passing through a quantum dot is investigated by using scattering theory. The transmission phase versus the intradot level shows a series of phonon-induced dips. These dips are highly sensitive to the electron-phonon interaction strength $\ensuremath{\lambda}$, and they are much more pronounced than phonon-assisted subpeaks appearing in the conductance. Phonon-induced dephasing is also studied, and the results show that the dephasing probability ${T}_{d}$ monotonically increases with the electron-phonon interaction strength $\ensuremath{\lambda}$. The dephasing probability ${T}_{d}\ensuremath{\propto}{\ensuremath{\lambda}}^{2}$ for small $\ensuremath{\lambda}$ but ${T}_{d}\ensuremath{\propto}\ensuremath{\lambda}$ at large $\ensuremath{\lambda}$.