This paper considers an intelligent reflecting surface(IRS)-aided wireless powered communication network (WPCN), where devices first harvest energy from a power station (PS) in the downlink (DL) and then transmit information using non-orthogonal multiple access (NOMA) to a data sink in the uplink (UL). However, most existing works on WPCNs adopted the simplified linear energy-harvesting model and also cannot guarantee strict user quality-of-service requirements. To address these issues, we aim to minimize the total transmit energy consumption at the PS by jointly optimizing the resource allocation and IRS phase shifts over time, subject to the minimum throughput requirements of all devices. The formulated problem is decomposed into two subproblems, and solved iteratively in an alternative manner by employing difference of convex functions programming, successive convex approximation, and penalty-based algorithm. Numerical results demonstrate the significant performance gains achieved by the proposed algorithm over benchmark schemes and reveal the benefits of integrating IRS into WPCNs. In particular, employing different IRS phase shifts over UL and DL outperforms the case with static IRS beamforming.