This paper investigates robust and secure multiuser multiple-input single-output (MISO) downlink communications assisted by a self-sustainable intelligent reflection surface (IRS), which can simultaneously reflect and harvest energy from the received signals. We study the joint design of beamformers at an access point (AP) and the phase shifts as well as the energy harvesting schedule at the IRS for maximizing the system sum-rate. The design is formulated as a non-convex optimization problem taking into account the wireless energy harvesting capability of IRS elements, secure communications, and the robustness against the impact of channel state information (CSI) imperfection. Subsequently, we propose a computationally-efficient iterative algorithm to obtain a suboptimal solution to the design problem. In each iteration, S-procedure and the successive convex approximation are adopted to handle the intermediate optimization problem. Our simulation results unveil that: 1) there is a non-trivial trade-off between the system sum-rate and the self-sustainability of the IRS; 2) the performance gain achieved by the proposed scheme is saturated with a large number of energy harvesting IRS elements; 3) an IRS equipped with small bit-resolution discrete phase shifters is sufficient to achieve a considerable system sum-rate of the ideal case with continuous phase shifts.