Try updating your Java: Java SE Downloads
Download the JRE offline installer & remove any other Java versions before installing this one.

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