Does your computer have to be on in order for home sharing to work?

Yes

That largely depends on what you are trying to do. If you want to view content from the iTunes library on your computer then yes the computer needs to be on. However if you are wanting to use the remote app on an iOS device, that utilises homesharing, then no the computer does not have to be on.

The computer must be on AND the iTunes application must be running.

I believe this matter needs clarifying.

The computer DOES NOT need to be on for homesharing to work.

Obviously if the computer isn't on then any services on the Apple TV which rely on homesharing to the computer, such as playing content from the iTunes library will not work, but homesharing with any other device WILL continue to work.