This paper studies the output containment control problem for heterogeneous multi-agent systems subject to unknown time-varying disturbances by designing fully distributed anti-disturbance control protocols. Fully distributed observers are designed to estimate convex combination of leader states, which are independent of any global information including global leaders' dynamics and global graph. There are external disturbances to contaminate the control inputs of the followers, which may damage the cooperation of the systems. Then adaptive anti-disturbance compensators are designed to eliminate the adverse effects of the external disturbances. Based on the proposed fully distributed observers and adaptive anti-disturbance compensators, adaptive anti-disturbance containment control protocols are provided to ensure that the output trajectories of each follower can converge to the dynamic convex hull spanned by the leader outputs. Finally, a numerical example and a practical example of heterogeneous mobile vehicles are given to demonstrate the effectiveness of the proposed method.