Mozilla has added support for asm.js to Firefox 22 Nightly
Mozilla is already taking advantage of this system with what it calls OdinMonkey, its implementation of asm.js.
OdinMonkey is now shipped with Firefox 22 Nightly, which will reach the stable channel in June. OdinMonkey should be enabled by default in Firefox 22 at that time, if everything goes according to plan.
The parallel with Google's Native Client is not coincidental, but Mozilla's approach has some big advantages. Here's how asm.js and OdinMonkey work in greater detail.
For the moment, only the Windows and Firefox builds support OdinMonkey. However, the Mac OS X builds are almost ready and so is the Firefox for Android ARM build. They should be ready by the time Firefox 22 ships.
You can check out the performance improvements for yourself by grabbing Firefox 22 Nightly and testing it with BananaBench. You should run it on another Firefox version first and then compare the results.