Neither in Ubuntu nor in Debian the tool visualvm is part of the OpenJDK 7 package. It's part of the Oracle JDK 7 and seems to be GPL licensed. While in Ubuntu it can be installed with a separate package, such a package doesn't exist in Debian.
Why isn't it part of the OpenJDK packages?
Our Ubuntu machine did not include VisualVM as part of the Java JDK. A quick search showed that the package was available as a separate install:
$ apt-cache search visualvm
...
visualvm - All-in-One Java Troubleshooting Tool
Quick Install:
$ apt-get install visualvm
After successful installation:
$ which jvisualvm
/usr/bin/jvisualvm
I tried running it, but it kept complaining that it was using the JRE. I edited /etc/visualvm/visualvm.conf
# default location of JDK/JRE, can be overridden by using --jdkhome <dir> switch
jdkhome="/usr/lib/jvm/java-6-openjdk-amd64"