Millimeter wave (mmWave) technology operates at high frequencies to provide dramatically increased bandwidth for next-generation networks. mmWave multiple-input multiple-output (MIMO) systems require hybrid precoding - a combination of digital and RF techniques - rather than purely digital precoding due to hardware constraints. This approach balances performance with cost while addressing limitations of signal mixers and analog-to-digital converters. These systems operate in wideband channels with frequency selectivity, requiring OFDM (orthogonal frequency-division multiplexing) to handle dispersive channels. We study the problem of robust digital-RF precoding optimization for the downlink sum-rate maximization in hybrid multi-user (MU) MIMO-OFDM systems under maximum transmit power and unit modulus constraints, where phase shifters (PSs) are possibly impaired at the transmitter and users. The formulated maximization problem is non-convex and difficult to solve. We propose a weighted minimum mean squared error (WMMSE) based block coordinate descent (BCD) method to iteratively optimize digital-RF precoders at the transmitter and digital-RF combiners at the users. Low-cost and scalable optimization approaches are proposed to efficiently solve the BCD subproblems. Extensive simulation results are conducted to demonstrate the efficiency of the proposed approaches and exhibit their superiority relative to well-known benchmarks.