Chris, thanks for the regression window, that was very helpful for confirming my theory.
TenFourFox doesn't officially support WebRTC or its components, including RTCPeerConnection. This is documented in the release notes. WebRTC is the underlying media conferencing framework that Google Voice uses. Earlier versions sort of worked, and we still build it as part of the browser, but it is unsupported due to certain features which are not compatible with 10.4 and 10.5 and video generally runs poorly which is its primary use case. It can be used for audio alone, however, which is what Google Voice uses it for.
I am not a Google Voice subscriber, but I was also able to confirm the problem (it also happens on the signup screen). In a debug version, this appears:
Officially this is not a bug I plan to fix, since I don't support this mode of operation, and there are likely to be other problems with Google Voice.
Unofficially, you can turn peer connections back on by going to about:config and setting media.peerconnection.enabled to true (you may need to restart the browser for this setting to stick). When I did this, the spinning circle disappeared and I was asked to sign up. I did not do further testing since I don't subscribe to Google Voice. If you find other bugs and you are interested in fixing this, I certainly do accept code submissions, though WebRTC will remain officially unsupported as a matter of policy.