The tectonic plates stopping per se would not be catastrophic. It would reduce the formation of new crust on volcanic vents and stop the formation of mountains. Eventually, the Earth would erode until almost all land was at the same level, meaning everything would be underwater. Life could definitely still survive that, though; indeed, most of the world's life lives underwater. Moreover, it would take millions of years to do this, so land-dwellers would have a lot of time to prepare. In the meantime, there would be no more earthquakes to deal with!
The real catastrophe would come from why the tectonic plates stopped moving. The only really plausible way this could happen would be if the Earth's core cooled and solidified, meaning there was no longer hot magma being fed upward into and through the mantle. If that ever happened, it would be catastrophic, though it would take a long time.
While the flow of the core—and the resulting magnetic field of the Earth—has changed direction and strength over thousands of years, it has never completely stopped. If it did, solar radiation reaching the surface would increase dramatically. This would have a direct effect of killing many organisms on the surface (it would be like the whole world was sitting in the Van Allen Belt), but the now-freed solar wind stripping away the Earth's atmosphere would be even more problematic. It would take a few billion years to strip away completely, and life might be able to adapt, but we're pretty sure this is what happened on Mars and that's why no life has been discovered there.