I'm guessing you want to know what is the minimum it will run on. Not how much it needs - slightly different question.
I'd guess that the minimum it would run on would be a Celeron 1.6ghz, which is about the lowest you could buy today.
Limewire is not a CPU heavy program, it could run on just about anything.
And just incase you meant the other question - well, like any program, it will use as much CPU as it needs. Programs are generally flexible. Limewire is likely more so, because it probably knows that it is usually a background process. So anything from 1 to 100%