We observe a giant increase of the infrared intensity and a softening of the in-plane antisymmetric phonon mode E(u) ( approximately 0.2 eV) in bilayer graphene as a function of the gate-induced doping. The phonon peak has a pronounced Fano-like asymmetry. We suggest that the intensity growth and the softening originate from the coupling of the phonon mode to the narrow electronic transition between parallel bands of the same character, while the asymmetry is due to the interaction with the continuum of transitions between the lowest hole and electron bands. The growth of the peak can be interpreted as a "charged-phonon" effect observed previously in organic chain conductors and doped fullerenes, which can be tuned in graphene with the gate voltage.