Infrared spectroscopy reveals unusual tunable electronic structure and optical behaviour in electrically gated bilayer graphene. In a dual-gate bilayer graphene device, we were able to control the carrier doping and a semiconductor bandgap independently by using different combinations of the top and bottom gate voltages. The field-induced bandgap can be probed directly through the emerging interband transitions in infrared absorption spectra. A tunable bandgap up to 250 meV has been observed in our dual-gate bilayer graphene devices. This unique tunable bandgap can lead to many new physical phenomena. One example is an unusual phonon–exciton Fano resonance when the electronic bandgap is tuned to match the phonon vibration energy. Here (continuous) electron–hole transitions and (discrete) phonon vibrations form a coupled system described by the Fano resonance, and the infrared absorption spectra exhibit characteristic quantum interference between the phonon and exciton transitions. Remarkably, this coupled phonon–exciton Fano resonance can be continuously tuned through electrical gating in bilayer graphene, and its behaviour is described quantitatively by theory.