Fano resonances are features in absorption, scattering or transport spectra resulting from the interaction of discrete and continuum states. They have been observed in a variety of systems. Here, we report a many-body Fano resonance in bilayer graphene that is continuously tunable by means of electrical gating. Discrete phonons and continuous exciton (electron-hole pair) transitions are coupled by electron-phonon interactions, yielding a new hybrid phonon-exciton excited state. It may also be possible to control the phonon-exciton coupling with an optical field. This tunable phonon-exciton system could allow novel applications such as phonon lasers.