Massive full-dimensional multiple-input multiple-output (FD-MIMO) base stations (BSs) have the potential to bring multiplexing and coverage gains by means of three-dimensional (3D) beamforming. The key technical challenges for their deployment include the presence of limited-resolution front ends and the acquisition of channel state information (CSI) at the BSs. This paper investigates the use of FD-MIMO BSs to provide simultaneously high-rate data communication and mobile 3D positioning in the downlink. The analysis concentrates on the problem of beamforming design by accounting for imperfect CSI acquisition via time division duplex-based training and for the finite resolution of analog-to-digital converter and digital-to-analog converter at BSs. Both unstructured beamforming and a low-complexity Kronecker beamforming solution are considered, where for the latter the beamforming vectors are decomposed into separate azimuth and elevation components. The proposed algorithmic solutions are based on the Bussgang theorem, rank-relaxation, and successive convex approximation (SCA) methods. Comprehensive numerical results demonstrate that the proposed schemes can effectively cater to both data communication and positioning services, providing only minor performance degradations compared to the conventional cases in which either only the data communication or only positioning is implemented. Moreover, the proposed low-complexity Kronecker beamforming is seen to guarantee a limited performance loss in the presence of a large number of BS antennas.