A fundamental problem in energy harvesting Wireless Sensor Networks (WSNs) is to maximize coverage, whereby the goal is to capture events of interest that occur in one or more target areas. To this end, this paper addresses the problem of maximizing network lifetime whilst ensuring all targets are monitored continuously by at least one sensor node. Specifically, we will address the Distributed Maximum Lifetime Coverage with Energy Harvesting (DMLC-EH) problem. The objective is to determine a distributed algorithm that allows sensor nodes to form a minimal set cover using local information whilst minimizing missed recharging opportunities. We propose an eligibility test that ensures the sensor nodes with higher energy volunteer to monitor targets. After that, we propose a Maximum Energy Protection (MEP) protocol that places an on-duty node with low energy to sleep while maintaining complete targets coverage. Our results show MEP increases network lifetime by 30% and has 10% less redundancy as compared to two similar algorithms developed for finite battery WSNs.