While DRI3 support is now quite stable across drivers if using relatively new versions of all the relevant software components (xorg-server, Mesa, the DDX drivers, etc), the main Radeon and Intel DDX drivers don't yet enable DRI3 by default but opt for DRI2. However, as shown in past tests, with the right conditions DRI3 can be noticeably faster while also having the benefit of more perfect rendering (a.k.a. hopefully no screen tearing).
It's easy at least to enable DRI3 via the xorg.conf/xorg.conf.d and some distributions like Fedora 23 have even enabled DRI3 by default in their packaged drivers. As some complementary data to the big Ubuntu 16.04 Radeon tests from earlier this week, I took three of the graphics cards and did an extra run when using DRI3 in place of the default DRI2.
The Ubuntu 16.04 system as mentioned in that article was using the Linux 4.4 kernel (but Canonical back-ported the Radeon changes from Linux 4.5) and the x-staging PPA with Mesa 11.2 built against LLVM 3.8.
The tested graphics cards with this extra DRI3 run were the R9 290, R7 370, and R9 Fury.
The test benefiting the most on this system from DRI3 was just OpenArena.
You can find all of this test data via this OpenBenchmarking.org result file. While this batch of DRI3 results may not be too exciting, the good news is that I didn't run into any issues or regressions when opting for Direct Rendering Infrastructure 3.
On a related note, over in this OpenBenchmarking.org result file I also ran some Intel DRI3 tests atop Ubuntu 16.04.