This specifically applies to Pale Moon 25.0 in Firefox Compatibility mode, and to any version prior to 25.0.
Cause:
If you are getting nagged by Google that "you are using a Firefox browser that is no longer supported" then that means Google thinks you are using Firefox 24 and they shove Pale Moon on the heap of "outdated/obsolete" browsers. The error lies with Google's "browser detection" script, and you should complain to them that they are giving you an error about Firefox when you are not using Firefox, but an up-to-date alternative.
Pale Moon is fully capable otherwise of displaying Google assets. This is just Google doing an arbitrary and inaccurate check on browsers that visit, and nagging users with "your browser is too old" messages in all their web services (GMail, Docs, etc.).
Although the premise behind giving a message like that is acceptable (not wanting to provide customer support for obsolete browsers), the execution is really poor. Their detection is flawed, and they pop up the message every single time, nagging the hell out of their users and causing a lot of annoyance. I'm sure people are aware of the situation after the first time

Workarounds:
You can disable "Firefox compatibility mode" in Options -> Advanced to avoid the nagging for the time being, but that in turn will make Google show the "unknown browser" compatibility layout (which is slightly less functional).
Alternatively, you can make a site-specific override for google.com (and whichever localized version you use of the domain) by following the instructions in this other FAQ entry.
Solution:
I'm updating the version presented to websites in the next version (25.1) of the browser (in line with HTML renderer updates) to prevent this kind of annoyance in Firefox Compatibility mode on Google and other sites, but please understand it's combating a symptom, not a cause.
With Firefox Compatibility mode disabled, Google will continue to show the slightly less functional "unknown browser compatibility view" layout, because that is simply something they need to fix by recognizing Pale Moon as a browser in their detection script, or using actual feature detection instead - the proper way to do this kind of thing for current-day web design.