The Internet of Things (IoT) will significantly change both industrial manufacturing and our daily lives. Data collection and 3-D positioning of IoT devices are two indispensable services of such networks. However, in conventional networks, only terrestrial base stations (BSs) are used to provide these two services. On the one hand, this leads to high energy consumption for devices transmitting at cell edges. On the other hand, terrestrial BSs are relatively close in height, resulting in poor performance of device positioning in elevation. Due to their high maneuverability and flexible deployment, unmanned aerial vehicles (UAVs) could be a promising technology to overcome the above shortcomings. In this article, we propose a novel UAV-assisted IoT network, in which a low-altitude UAV platform is employed as both a mobile data collector and an aerial anchor node to assist terrestrial BSs in data collection and device positioning. We aim to minimize the maximum energy consumption of all devices by jointly optimizing the UAV trajectory and devices' transmission schedule over time, while ensuring the reliability of data collection and required 3-D positioning performance. This formulation is a mixed-integer nonconvex optimization problem, and an efficient differential evolution (DE)-based method is proposed for solving it. Numerical results demonstrate that the proposed network and the optimization method achieve significant performance gains in both energy-efficient data collection and 3-D device positioning, as compared with a conventional terrestrial IoT network.