Does my computer need to stay on for PhoneTray to work?

Your computer has to be powered on (not in standby / sleep mode) in order for PhoneTray to work. PhoneTray cannot block unwanted calls when you computer is turned off. You can configure your PC to turn the display off but sleep mode has to be disabled (set to never).

To adjust power settings for your PC, open Control Panel -> Hardware and Sound Options -> Power Options and click Change when the computer sleeps on the left panel. Change Put the computer to sleep option to Never:

Even with the sleep option set to Never, you can still put your computer to sleep or turn it off manually using Shut down / Sleep / Hybernate options in Windows Start menu.