We propose a tunable coherent perfect absorber based on ultrathin nonlinear metasurfaces. The nonlinear metasurface is made of plasmonic nanoantennas coupled to an epsilon-near-zero material with a large optical nonlinearity. The coherent perfect absorption is achieved by controlling the relative phases of the input beams. We show that the optical response of the nonlinear metasurface can be tuned from a complete to a partial absorption by changing the intensity of the pump beam. The proposed nonlinear metasurface can be used to design optically tunable thermal emitters, modulators, and sensors.