Often when I return to the home screen from using an app, the HTC Sense UI (running 2.1) restarts. It seems to be worse when more apps are running. Does it shutdown to free memory, is there a stability problem or something else?
Whenever android needs more ram, it will kill anything in the "background" the HTC sense is background if you are playing a video game or any other app. So when android sees that it needs ram, sense with it's fairly large memory footprint is a prime candidate. Think nothing of it. If it crashed you would get a "force close". Some launchers do tricks to force them to not be killed by android BUT that has a negative effect on ram availability for foreground apps.
Most likely you have too many services running at once that are eating up all of your memory.
Check running services under Settings --> Applications --> Running services. Look at the colorful memory bar at the bottom of how much free memory there is.
Look for services that are high on memory usage and that you don't use anymore. Ask yourself if you really need that service running or not. If you don't, remove it. See if that helps. You'll probably see a better performance boost everywhere if you are being bogged down by excessive amounts of services.
Additionally, you could try out a new Home launcher such as LauncherPro.