I hadn't that problem with Mathematica v10.0.2 but I have it now with the last Mathematica v10.1.0 (Windows 8.1/Firefox 36.0.4).
It seems that Mathematica 10.1.0 has installed a new plugin npmathplugin.dll in the folder
*c:\Program Files (x86)\Common Files\Wolfram Research\Browser\10.1.0.5292844* and that's the one reported by Firefox (about:plugins):
Path: C:\Program Files (x86)\Common Files\Wolfram Research\Browser\10.1.0.5292844\npmathplugin.dll
I've tried to change the npmathplugin.dll file with the one that was installed by the previous Mathematica v.10.0.2, fished out from a recent HD backup. The folder was then *C:\Program Files (x86)\Common Files\Wolfram Research\Browser\10.0.2.5203600* and it was removed with the installation of v. 10.1.0.
The result is that the previous npmathplugin.dll (the one coming with v.10.0.2) works again in Firefox (i.e. I can load and see in the browser the Demonstrations of Wolfram Demonstrations Project).
So I suppose that Mathematica 10.1.0 installs a browser plugin that's not actually compatible with Firefox.
One thing to try could be uninstall the Mathematica Extras coming with Mathematica 10.1.0 and maybe reinstall the last CDF Player (it should restore the previous plugin).
The working npmathplugin.dll has size 4285720, while the not working npmathplugin.dll has size 4285744
EDIT - UPDATE
After restoring the npmathplugin.dll file (the one installed by Mathematica 10.1.0) and checking again the plugin works again, so all I wrote before doesn't actually apply.
That's sort of a mistery. Maybe the problem is in Firefox cache. I'm very puzzled and confused.