There is a mirror on dmoztools.net. However, I don't know if it's an exact replica. Obviously, the Wayback Machine has a lot of copies of Dmoz, but I don't know how deep it went. I know Dmoz used to let you download their entire dataset/database. Did they publish a final version of their dataset/database before they closed?
I can't post the link to the wayback machine, but it's on archive.org if you've never used it. The dmoztools site also doesn't have it's own internal search engine, so you have to use a Google site search using the site:dmztools.net parameters. There's also no contact info listed on the site, so I don't know who runs it, and it points back here for discussion, but I didn't realize this is the official discussion for the now defunct version of dmoz.org.
UPDATE: I found another dmoz mirror at -.com. It looks a bit more professional, but just like back in the day, anyone could run a dmoz clone, just like now, as anyone can run a wikipedia clone, they could just be using it as a money making site. Nonetheless, now that Dmoz is down, there are probably a lot of mirrors around out there. But, it's hard to tell if they got a copy of the final version of dmoz before it closed.
I think you may still be able to download the last RDF from Archive.org, if you can remember where DMOZ stored it. DmozTools was started by an editor, and seems to be a semi-official mirror til there's something editable.
Do you know the name of the editor or how to get in contact with them? I just wanted to let them know my suggestion about adding a search engine on the site. I did a WHOIS on the domain, but it's protected by a privacy service. I emailed the privacy service email anyway, as I think it forwards the email to the real contact, but I've never gotten a reply from one of those, ever.
Dmoztools is at the moment the official continuation of dmoz, and is provided by the editor community. Work is ongoing on the new site. I do not think there is any need for external suggestions on the functionality of the site.
@pogue - If you see something seriously not right with Dmoztools.net, by all means report it in Bugs or Quality Control - But: As informator said above, it is a non-editable directory mirror and normally won't have more features added.
Both Jasmine and Curlz are projects belonging to individual editors, and aren't officially part of the editing community. @stillbuyvhs - If you are looking for a directory data download, Curlz provides this function (linked here) but it is the last RDF produced by Dmoz and as such, no longer updated.
You aren't the only one curious to see what they'll do next, but that is out of our hands now; we are concentrating on launching a replacement editable directory... @David Kiley - PM me your editor name, starting category and approximate number of edits made; granting you editor status here is a part of rejoining you to Curlie...