We show that terrestrial planets in the habitable zones of M dwarfs older than $\sim$ 1 Gyr could have been in runaway greenhouses for several hundred Myr following their formation due to the star's extended pre-main sequence phase, provided they form with abundant surface water. Such prolonged runaway greenhouses can lead to planetary evolution divergent from that of Earth. During this early runaway phase, photolysis of water vapor and hydrogen/oxygen escape to space can lead to the loss of several Earth oceans of water from planets throughout the habitable zone, regardless of whether the escape is energy-limited or diffusion-limited. We find that the amount of water lost scales with the planet mass, since the diffusion-limited hydrogen escape flux is proportional to the planet surface gravity. In addition to undergoing potential desiccation, planets with inefficient oxygen sinks at the surface may build up hundreds to thousands of bars of abiotically produced O$_2$, resulting in potential false positives for life. The amount of O$_2$ that builds up also scales with the planet mass; we find that O$_2$ builds up at a constant rate that is controlled by diffusion: $\sim$ 5 bars/Myr on Earth-mass planets and up to $\sim$ 25 bars/Myr on super-Earths. As a result, some recently discovered super-Earths in the habitable zone such as GJ 667Cc could have built up as many as 2000 bars of O$_2$ due to the loss of up to 10 Earth oceans of water. The fate of a given planet strongly depends on the extreme ultraviolet flux, the duration of the runaway regime, the initial water content, and the rate at which oxygen is absorbed by the surface. In general, we find that the initial phase of high luminosity may compromise the habitability of many terrestrial planets orbiting low mass stars.