I just upgraded to Mac OS X 10.7, and DrJava works flawlessly. The one vague report that someone had filed  therefore proved to be wrong.
When I started DrJava the first time, Mac OS X asked me if I wanted to
download a Java runtime environment. I said yes. The JRE was
downloaded and installed, and after that, DrJava opened.
I’m still not sure about Mac OS X Lion, but I continue to be excited about DrJava: