Disk space just sits there, it doesn't do anything. It can't slow the server down unless it actually uses it.
I take it you have a performance problem in your server.
If it's a straightforward problem of the hardware being simply insufficient, you still have to figure out which straightforward problem it is.
Not enough CPU power? Faster CPU or, if the application and hardware supports it, more CPU cores. Not enough disk speed? Faster disks, or more disks RAIDed in a more parallel manner, or reconfigure your filesystem/application to work more optimally. Not enough memory? More memory.
It might
not be a hardware problem for all we know, though. Can you give us more details on what's happening?