The cost and power consumption associated with the large number of linear power amplifiers required at the base station in a massive MIMO-OFDM system present a significant challenge when deploying systems commercially. In this paper, we show that capacities of up to 4 bits per channel use (bpcu) per user can be achieved in the massive MIMO downlink whilstonly transmitting the phase component of the precoded OFDM signals. This significantly reduces the peak-to-average power ratio (PAPR) of the transmit signals and relaxes the peak power and linearity requirements of the power amplifiers. Using Bussgang's theorem, it is shown that discarding the envelope information causes each OFDM subcarrier to be corrupted by an uncorrelated additive Gaussian error, which is received non-coherently by the users and asymptotically disappears as the number of transmit antennas is increased. SINR expressions are derived for linear precoders, and Monte Carlo simulations are performed with both Rayleigh and measured massive MIMO propagation channels. Under practical numbers of transmit antennas, we show that this phase-only OFDM scheme gives similar performance to other constant envelope precoding schemes described in the literature, and can achieve capacities of 2 bpcu per user with a 1-2 dB increase in mean transmit power (or 50% more transmit antennas) and 4-5 dB decrease in peak transmit power, compared to standard OFDM.