This almost already exists...
See, the idea is that a question that gets closed will eventually be deleted (or re-opened). It's a sort of "limbo", where questions sit until the community decides on their fate. But there's almost no value in having hundreds of closed questions littering up the site, with the very notable exception of duplicates...
Once a question is deleted, any reputation points gained by the folks answering it is forfeit - they won't lose it immediately, but the next time that user, a moderator, or a developer triggers a reputation recalc, it's gone.
So the real lesson here is the story isn't over just because a question gets closed: sooner or later, someone needs to step up and finish the job:
Moderators can delete anything. So feel free to ask them to delete a question that's sitting around stinking up the place.
10K users can vote to delete closed questions after they've been closed for two days. So if you have at least 10K reputation points on the site, feel free to go through the list of closed questions and vote to delete stuff.
20K users can also vote to delete, but don't have to wait two days. So if you've managed to rack up 20K reputation points, go nuts.
But don't delete duplicates
Seriously, don't. Unless they're literally word-for-word duplicates, just leave them be - folks tend to find all sorts of different ways of asking the same questions, and the more variations we have pointing to the same canonical answer, the better. Of course, you should
Merge duplicates with good answers
As Mitch notes, it's easy to answer a duplicate without realizing that it is a duplicate. As the site grows, this gets easier and easier. An honorable person might, upon realizing their answer adds nothing to what's already provided elsewhere, delete their own answer... But it's also common for duplicates to get better answers than the original - in these cases, asking a moderator to step in and merge the two ensures the answers get collected under the original question.