This paper studies intelligent reflecting surface (IRS)-assisted wireless-powered communication networks (WPCNs), where a hybrid access point (HAP) broadcasts energy signals to multiple devices for their energy harvesting in the downlink (DL) and then the devices use the harvested energy to transmit information signals to the HAP in the uplink (UL) with the help of an IRS. We adopt a practical non-linear energy harvesting (EH) model and propose a power-splitting (PS) EH receiver architecture with multiple rectifiers to avoid the input radio-frequency power to get stuck into the saturation regime. To fully unleash the potential of IRS, we propose a dynamic IRS beamforming design, where the IRS phase-shift vectors vary across the durations of DL wireless energy transfer (WET) and UL wireless information transmission (WIT). The objective of this paper is to minimize the transmit energy consumption at the HAP by jointly optimizing the DL/UL time allocation, the HAP/devices transmit power, the PS factor, and IRS phase shifts, subject to a set of minimum throughput requirements for individual devices. To address the resulting non-convex optimization problem, an efficient alternating optimization based on the successive convex approximation (SCA) technique is proposed. Simulation results demonstrate the effectiveness of our proposed design over various benchmark schemes and also unveil the importance of the joint design of IRS beamforming and PS rectifiers for achieving energy efficient WPCNs in practice.