Try updating your Java: Java Downloads for All Operating Systems - Sun Microsystems
Download the offline installer & remove any other Java versions (via Control Panel) before installing this one. Restart your computer after installing the new Java.

Next, delete LW's preferences folder when LW's properly closed. See point 3 here & also point 2 if you've used older versions of LW in the past (there could be a folder in either location).
Fixes for LimeWire
This will change all of your settings within options, you'll have to go back through afterwards & reset things. If you're not sure of your settings, go through LW's tools > options & write them all down, so that you can reset things later.
Vista users: LimeWire Pref Folder in Vista

If that doesn't help, right click on the LW icon > properties > compatibility. Try running in compatibility mode for Win 2000 or NT. Vista users, right click the LW icon & run as adminstrator & also try compatibility mode for XP.

Do you have Alcohol software installed? If so, look here.
LW starts then close, log: resolve_virtual_call

You could also try the advice here: FIX: LimeWire won't start or closes immediately