No, shortcuts on a stand-alone computer will not degrade performance. If you want to tidy your desktop, you can create folders and group your shortcuts in the folders (e.g. all office-type programs in one, net, email etc in another, video in a third etc)
In big networks, particularly when you can log on to any computer, your personal desktop is held on the central server and downloaded to the computer that you log on to. If you have lots of shortcuts, etc, it takes up more bandwidth and time to download them.