Intelligent reflecting surface (IRS) and movable antenna (MA) technologies have been proposed to enhance wireless communications by creating favorable channel conditions. This paper investigates the joint beamforming and antenna position optimization for MA-enabled IRS (MA-IRS)-aided multi-user multiple-input single-output (MU-MISO) communication systems, where the MA-IRS is deployed to aid the communication between the MA-enabled base station (BS) and user equipment (UE). In contrast to conventional fixed position antenna (FPA)-enabled IRS (FPA-IRS), the positions of the reflecting elements of the MA-IRS can be controlled to enhances the wireless channel. To verify the system's effectiveness and optimize its performance, we formulate a sum-rate maximization problem with a minimum rate threshold constraint for the MU-MISO communication. To tackle the non-convex problem, a product Riemannian manifold optimization (PRMO) method is proposed for the joint optimization of the beamforming and MA positions. Specifically, a product Riemannian manifold space (PRMS) is constructed and the corresponding Riemannian gradient is derived for updating the variables, and the Riemannian exact penalty (REP) method and a Riemannian Broyden-Fletcher-Goldfarb-Shanno (RBFGS) algorithm is exploited to obtain a feasible solution over the PRMS. Simulation results demonstrate that compared with the conventional FPA-IRS-aided communications, the reflecting elements of the MA-IRS can move to the positions with higher channel gain, thus enhancing the system performance. Furthermore, it is shown that optimizing the positions of the reflecting elements brings higher performance gain than controlling the phase shifts of the IRS, and integrating MA with IRS leads to higher performance gains compared to integrating MA with BS.