The trade-off between the achievable sum rate and harvested energy is investigated for multi-user/multi-cell massive multiple-input multiple-output (MIMO) relay networks. To this end, the antenna array at the relay is switched periodically between the energy harvesting and information processing modes, and the harvested energy is utilized for the relay's transmissions. Closed-form expressions for the asymptotic signal-to-interference-plus-noise ratio, achievable sum rate and harvested energy at the relay are derived whenever the numbers of antennas at the relay and destination grow without bound. This analysis reveals that the presence of co-channel interference (CCI) can be exploited to significantly boost the harvested energy at the massive MIMO relay. However, the detrimental effects of CCI for signal decoding can effectively be mitigated by using a zero-forcing linear detector at the destination in the limit of infinitely many relay and destination antennas.