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):

 Wolfram Mathematica
    File: npmathplugin.dll
    Path: C:\Program Files (x86)\Common Files\Wolfram Research\Browser\10.1.0.5292844\npmathplugin.dll
    Version: 10.1.30746.62445

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.

When did you download the CDF Player installer? Two weeks ago a Firefox-compatible CDF Player installer went into the User Portal for downloading.

Is Firefox running in 32-bit or 64-bit mode? The plug-in needs 32-bit. See
https://support.mozilla.org/en-US/questions/1028263

crash reporter says:

Is there a crash report generated on windows for CDF crash? Can you please show where it is located? or how did you find the CDF crashed? thanks

I have no problem with FF 35.0.1 and I see the same version number of the CDF plugin. Is the plugin enabled in FF Add-ons Manager-Plugins?

Supposing you are on Windows, have you tried quitting Mathematica through the small icon in the notification area? I think that the plugin doesn't work if there's some Mathematica kernel active (or launched by the Mathematica program or, maybe, left running from some other browser).