Submitted by Jack Collins, Duarte, California, and answered by Kerry Sieh, Professor of Geology, Caltech.
If all plate motion stopped, Earth would be a very different place. The agent responsible for most mountains as well as volcanoes is plate tectonics, so much of the activity that pushes up new mountain ranges and creates new land from volcanic explosions would be no more. The volcanoes of the Pacific Ring of Fire, in South and North America, Japan, the Philippines, and New Zealand, for example, would shut off, and the steady southeastward migration of volcanic activity along the Hawaiian Islands would stop. Volcanism would just continue on the big island. There would also be far fewer earthquakes, since most are due to motion of the plates.
Erosion would continue to wear the mountains down, but with no tectonic activity to refresh them, over a few million years they would erode down to low rolling hills. So the whole planet would be flatter, and the topography would be a heck of a lot less exciting. You'd probably be less inclined to go trekking in Nepal.
One big problem with plate tectonics stopping is that plate motion is the mechanism by which Earth is cooling down and getting rid of its internal heat. If the plates stopped moving, the planet would have to find a new and efficient means to blow off this heat. It's not clear what that mechanism might be.