An intelligent reflecting surface (IRS)-aided wireless-powered mobile edge computing (WP-MEC) system is conceived, where each device’s computational task can be divided into two parts for local computing and offloading to mobile edge computing (MEC) servers, respectively. Both time division multiple access (TDMA) and non-orthogonal multiple access (NOMA) schemes are considered for uplink (UL) offloading. To fully unleash the potential benefits of the IRS, employing multiple IRS beamforming (BF) patterns/vectors in the considered operating frame to create time-selectivity channels, i.e., dynamic IRS BF (DIBF), is in principle possible at the cost of additional signaling overhead. To strike a balance between the system performance and associated signalling overhead, we propose three cases of DIBF configurations based on the maximum number of IRS reconfiguration times. The degree-of-freedom provided by the IRS may introduce different impacts on the TDMA and NOMA-based UL offloading schemes. Thus, it is still fundamentally unknown which multiple access scheme is superior for MEC UL offloading by considering the impact of the IRS. To answer this question, we provide a comprehensively theoretical performance comparison for the TDMA and NOMA-based offloading schemes under the three cases of DIBF configurations by characterizing their achievable computation rate. Analytical results demonstrate that offloading adopting TDMA can achieve the same computation rate as that of NOMA, when all the devices share the same IRS BF vector during the UL offloading. By contrast, computation offloading exploiting TDMA outperforms NOMA, when the IRS BF vector can be flexibly adapted for UL offloading. Then, we propose computationally efficient algorithms by invoking alternating optimization for solving their associated computation rate maximization problems. Our numerical results demonstrate the significant performance gains achieved by the proposed designs over various benchmark schemes and also unveil that the optimal time allocated to downlink wireless power transfer can be effectively reduced with the aid of IRSs, which is beneficial for both the system’s spectral efficiency and its energy efficiency.