I cannot conceive of a legal environment in which such a project even has the barest breath of hope of happening.
Actually, it's not especially difficult legally. Jim absolutely can't read some random fanfic sent to him in an e-mail, no argument. But I don't think that's what Squidnurse is proposing. His idea, I think, can be more easily compared to that of a TV show.
Let's say you want to get a job writing for "Chuck" (first thing comes to mind). You can't just send a script to them, obviously; they can't read it, and you'd be lucky if they even opened it before trashing it or sending it right back to you. But, if you sign a release that says, basically, "I understand you could be working on something similar to what I'm sending you, and I really can't do a lot if you produce something like this later," (you can, of course, try, but it's an uphill battle) you can (though a lot of places don't really do that and you need an agent even to get to that point). Granted, most shows aren't reading scripts for their show, but it's somewhat less a legal concern and more a creative one, as the "Chuck" people are going to be a hell of a lot more picky about a "Chuck" script than they're going to be on one for... "Reaper".
So it's easy enough for Jim to be more or less legally secure in reading a story set in the Dresdenverse. In terms of the logistics of such a project, it's still not so completely unwieldy. Jim could furnish the equivalent of a TV show bible with the general rules of how magic works in the universe, some non-essential background on the Vampire and Fairy Courts, etc. (obviously, he couldn't throw out anything actually necessary to the story of the Dresden Files). So that's the general guide for submissions.
The problem arises in the spirit of the proposed project. Letting new writers get a foot in the door. I mean, that means a lot of unadulterated crap is going to get submitted. The simplest way to cull most of the crap herd is to require a short synopsis (one page should do). Many will likely ignore the rule and get cut by default while others will have to admit in about the first line that they've included Harry or Murphy or Kincaid as a main character (*snip*). After the easy cuts of the blatantly non-compliant, you take the best of the synopses (obviously, some perfectly compliant stories will still just not be good enough) and request the full stories. Which get cut again in a similar way (though, obviously fewer cuts for being completely off-base and more for just not being very good). Jim's publishers could get it down to, say, thirty of the best stories and submit those to him, and he'd make the final cut, possibly tweak or provide notes to get the stories into full compliance with canon, and you publish (ideally, paperback and halfway between main Harry releases).
The real question isn't legal, and it's not even really how to do it. It's a cost and creative question. Would the money and effort spent doing this make it worthwhile? And then, the creative question, would Jim even want something like this out there? If it turned out it'd make money and Jim were on board, then the legality of it is really fairly easy to work out.
In the interests of full disclosure (and explaining why such an old topic just popped up, I know Squid from another board, where this topic came up, and I felt compelled to throw in my two cents.