I'm not sure why the installer can't find the classes that are included in the installer. It should work. You could try setting an explicit CLASSPATH to the current directory just before you run the installer:
CLASSPATH=.; export CLASSPATH
Then run the installer:
sh LimeWireLinux.bin
I'm not sure that this will work though. If you can't find another solution you can try an alternative method of installing LimeWire:
http://www.cyberspace.org/~kohl/limewirelinuxhowto.html