As a product manager, I open a case with the feature design, R&D Manager opens multiple subcases and assigns them to developers. Each developer assigns that subcase to QA which test it and close the case once it is fixed.
I would like to get a notification when all those subcases are closed in order to know the feature is now ready.
The best way to handle this depends on how many non-Feature cases exist in your FogBugz installation.
You could use the Custom Workflow plugin to define a workflow in which cases get assigned back to you on resolution (which has to happen before closing a case). Alternatively, you could use the Auto-Subscribe notification, although this will trigger more notifications than you might want.
There's no way to do this with notifications, but just looking at your filter once a day, showing all your feature cases and all open children, should give you an idea of where things stand. Alternatively, you could use the API to loop through all feature cases and check to see if they have open children, then send an email when one is found that has no open children.