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.
If you're using Vista: LimeWire Pref Folder in Vista

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

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

Do you have AllChars installed? If you do, remove that & see whether LW opens.

What OS are you using? Was an earlier version of LW working before? Which version?