Compared with traditional half-duplex wireless systems, the application of emerging full-duplex (FD) technology can potentially double the system capacity theoretically. However, conventional techniques for suppressing self-interference (SI) adopted in FD systems require exceedingly high power consumption and expensive hardware. In this paper, we consider employing an intelligent reflecting surface (IRS) in the proximity of an FD base station (BS) to mitigate SI for simultaneously receiving data from uplink users and transmitting information to downlink users. The objective considered is to maximize the system weighted sum-rate by jointly optimizing the IRS phase shifts, the BS transmit beamformers, and the transmit power of the uplink users. To visualize the role of the IRS in SI cancellation, we first study a simple scenario with one downlink user and one uplink user. To address the formulated non-convex problem, a low-complexity algorithm based on successive convex approximation is proposed. For the more general case considering multiple downlink and uplink users, an efficient alternating optimization algorithm based on element-wise optimization is proposed. Numerical results demonstrate that the FD system with the proposed schemes can achieve a larger gain over the half-duplex system, and the IRS is able to achieve a balance between suppressing SI and providing beamforming gain.