Deauthorizing a working computer: does the iTunes library need to be present?
The authorization link is between the computer and the Apple ID, not any particular iTunes Library. So, no, you don't need the library to break that connection. You can create an empty library temporarily, just to get iTunes to launch, then use Deauthorize This Computer.

If you throw that computer in the garbage, you can always go to your iTunes account on another computer and click "Deauthorize All" to remove all authorizations at once.  You can then reauthorize your current computers as needed.