Try steps 1, 3 & 5 of the sticky below. When you update Java, make sure to use the JRE offline installer. Remove your current Java version before you install the new one.
Fixes for LimeWire

If that doesn't help, reinstall LW & use the offline installer (not the recommended one)
http://www.limewire.com/english/content/download.shtml

If it's still not opening after that, if you're using any antispyware, make sure that LW's on the list of exceptions/allowed apps. Some programs will wrongly pick LW as a threat & stop it from working properly.