This paper investigates the distributed fault-tolerant consensus problem for nonlinear multi-agent systems (MASs). A novel distributed fault-tolerant control protocol is proposed under a zero-sum differential game framework, where the consensus problem is reformulated as a minimax optimization between the control inputs of agents and the actuator faults through a local cost function. A critic neural network is trained online to solve the coupled Hamilton-Jacobi-Isaacs (HJI) equation, where the optimal control and upper bound of fault compensation are simultaneously derived from the Nash equilibrium condition. Leveraging the Lyapunov stability theorem, it is rigorously proved that the designed distributed fault-tolerant consensus control law guarantees the uniform ultimate boundedness (UUB) of the closed-loop systems. Simulation results validate the effectiveness of the present method.