We report measurements of the infrared optical response of thin black phosphorus under field-effect modulation. We interpret the observed spectral changes as a combination of an ambipolar Burstein-Moss (BM) shift of the absorption edge due to band-filling under gate control, and a quantum confined Franz-Keldysh (QCFK) effect, phenomena which have been proposed theoretically to occur for black phosphorus under an applied electric field. Distinct optical responses are observed depending on the flake thickness and starting carrier concentration. Transmission extinction modulation amplitudes of more than two percent are observed, suggesting the potential for use of black phosphorus as an active material in mid-infrared optoelectronic modulator applications.