I have a mac running on OS X Tiger, i have 652.8MB available. I had Lime wire installed on this very same computer a year and a half ago, it worked fine then. The initial download is fine but when i open it to install the icon bounces and disappears. I tired dragging it in to my applications folder but it does the same thing then too.

Any suggestions?

Do a safe boot & repair permissions, SAFE Boot & Repair Permissions. And if still no go then check options here Can't launch LW: Bouncing in Dock (click on link)

I have to say though, 650 MB is a dangerously low amount of HDD space to have left. All OSX programs including the OS itself use Virtual Memory (VM) which means they store memory on the HDD so you should really have several GB available for both the system & apps to behave properly. Can't you create more space somehow on the OSX drive?