In this paper, we propose a novel energy-efficient data collection and wireless power transfer (WPT) framework for internet of things (IoT) applications, via a multiple-input multiple-output (MIMO) full-duplex (FD) unmanned aerial vehicle (UAV). To exploit the benefits of UAV-enabled WPT and MIMO FD communications, we allow the MIMO FD UAV charge low-power IoT devices while at the same time collect data from them. With the aim of saving the total energy consumed at the UAV, we formulate an energy minimization problem by taking the FD hardware impairments, the number of uploaded data bits, and the energy harvesting causality into account. Due to the non-convexity of the problem in terms of UAV trajectory and transmit beamforming for WPT, tracking the global optimality is quite challenge. Alternatively, we find a local optimal point by implementing the proposed iterative search algorithm combining with successive convex approximation techniques. Numerical results show that the proposed approach can lead to superior performance compared with other benchmark schemes with low computational complexity and fast convergence.