I also had the same problem with LimeWire 1.6d for OS X. I decided to open the package contents, and under Contents/Resources I noticed that a line in MRJApp.properties was different in the beta than the 1.6b release. The line:

com.apple.mrj.application.classpath=/Users/limewire/Desktop/LimeWire6d/LimeWire.jar:/Users/limewire/Desktop/LimeWire6d/collections.jar

should be:

com.apple.mrj.application.classpath=Contents/Resources/Java/LimeWire.jar:Contents/Resources/Java/collections.jar

Everything worked after making that change. There was another line that was different:
com.apple.mrj.application.main=a.a.a.b.Main
is now
com.apple.mrj.application.main=com.limegroup.gnute lla.gui.Main

but I think that's how it should be. The line I changed; however, definitely seems to be the issue as it gives the path to the jar files and everything appears to work after making the change.

Oh, and one more thing. I tried posting this under my registration (iammanta), but it said I didn't have permission to post. I can post fine as Unregistered though. Interesting....

Hope this helps.