We theoretically study the effect of electron-phonon (e-ph) interaction on a Fano resonance in an Aharonov-Bohm interferometer with an embedded quantum dot. We examine the bias-voltage (V) dependence of the decoherence, using Keldysh Green function method and perturbation with respect to e-ph interaction. With optical phonons of energy ω0, only the elastic process takes place when eV < ω0, in which electrons emit and absorb phonons virtually. The process suppresses the resonant amplitude. When eV > ω0, the inelastic process is possible which is accompaniedby real emission of phonons. It results in the dephasing and broadens the resonant width. With acoustic phonons, the asymmetric shape of the Fano resonance grows like a symmetric one as the bias voltage increases, in qualitative accordance with experimental results.