Thermodynamically, a rotating planet and a non-rotating planet (with respect to their suns, of course) both behave the same way: they reach an equilibrium temperature and stay there, because as they heat up due to incoming radiation, their temperature causes them to radiate more energy away, until the outgoing radiation and incoming radiation balance. Of course, this is an average for the planet as a whole: for example, any given point on Mercury's surface is more or less in thermodynamic equilibrium with the half of the sky visible from that point, which will obviously be much higher temperature with the sun than without.
On any planet around a sun, rotating or not, there will be a gradient of temperature from one side to the other, and usually a polar gradient as well (that is, if the planet rotates and revolves along roughly the same plane). And within a certain range of distances from the sun, that gradient will include a zone where life as we know it is possible, given the right conditions. Those conditions include not only temperature but many others as well, primarily chemical in nature, such as the availability of water and the substances necessary for building organic molecules. Humans could live in the Twilight Zone of Mercury easily enough, for example, as long as we had access to food, water and air.
That's where you'd run into trouble. If Earth stopped spinning, for example, one aide of the Earth would heat up and the other would get very cold. Over time, the water that evaporated from the light side would snow onto the dark side and stay frozen there without any heat source to melt it. The weather between the two sides would be violent. Most life forms we are familiar with would not survive such an environment...
In the bright side, tardigrades would probably still make it.