The source code to the Doom 3 / id Tech 4 engine was released in November of 2011. Started immediately thereafter was the ioDoom3 project and backed by some of the same developers working on the thriving ioquake3 project that's powering many games. While it looked promising at first and there was hope open-source game projects would move to the newer engine or that new projects would come about, nothing has really happened.
The ioDoom3 GitHub repository hasn't seen any serious activity to the main repository since December of 2011. There was a commit one year ago to the day by Timothee Besset (the former id Software Linux developers) telling any developers cloning the code to find an active fork of the project.
Last week, the project fully migrated from Gitorious to GitHub, as announced on their blog. They are no longer maintaining their Gitorious repositories at all but will just be using GitHub. In their announcement of the GitHub change, the project concedes, "We recognize that not a whole lot has been happening with this project. Fortunately, Zakk informed me that after the next ioquake3 release, he plans to start pulling changes from your repositories!"
There also isn't any new developments within the upstream Doom3.gpl code repository or with the Doom 3 BFG source-code that id Software provided last year.
The last news on the matter was Doom 3 BFG being ported to Linux.
Aside from ioDoom3, sadly there is no longer much activity with the dhewm3 fork. There's also nothing with the Dante Engine due to the unfortunate passing of Oliver McFadden.
Hopefully more work will happen to the open-source Doom 3 code-base this summer, ideally with adoption by an interesting and/or noteworthy project.