I have not upgraded to Lion yet. Even though Lion is so cheap, the local reviews of people who had tried it was poor. Plus I am not keen on losing Rosetta which allows programs designed to run on PowerPC processors, to run on later OSX versions such as intel macs. Lion dropped the option for Rosetta. Thus I'd lose the ability to use several programs.

Are you booted into 32 or 64 bit OSX?

Here's one thing to try, Get Information for LimeWire (command i), check the option for running in 32 bit. I know my LW only runs properly on my Snow Leopard in this mode. Else like you, it crashes soon after opening.
Else if it is already checked, try unchecking the 32 bit option.

I notice a part of the error refers to Carbon. I didn't think Lion supported Carbon now. I am not too bright sometimes lol.

Have you checked for Java updates? One thought is you have upgraded Lion overtop of an earlier OSX. LW may run properly if on a fresh Lion installation with no prior OSX installation history.

Another thing to try is to do a SAFE boot (ie: hold down shift key as you boot until you see the apple face), then use Disk Utility to repair permissions. This is recommended after any OSX updates (or adobe for that matter.) . (On that topic, I hate Micro$oft & Adobe installing their junk all over the system.)