I'm also experiencing the same issue with my Finder on OS X 10.9. It started yesterday on my end too.
Finder windows that have recently been opened close themselves shortly after and the icons on the desktop disappear very briefly and then come back as if the Finder was restarting itself.
If somebody knows about this, we'd be grateful for a tip.
Using Google Drive? Here's your culprit. Go to the Google Drive preferences (right-click the menu bar icon) and disable "show file sync status icons". Found this solution in a similar discussion and works great for me: no more Finder restarts.