Try updating your Java: Java SE Downloads
Download the JRE offline installer & remove any other Java versions before installing this one.
If that doesn't help, go back to an earlier Java version: Archive: Download Java 2 Platform Standard Edition (J2SE) 5.0 Update 11
Same thing with the earlier version...download the JRE offline installer & remove all other Java versions before you install 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. Remember to show hidden files & folders so that you can find the preferences folder.
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

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

Even if you're not getting an error message, try these things anyway.
Which version of LW are you using? Have you been able to get LW connecting & opening in the past? Which version worked?