@ jmm
I only use FF and Avast never turns itself off. I think what you've been reading in the forum is that some people have had an issue with Avast disabling itself regardless of what browser they use and that's when they are seeing the red "X" on top of the orange ball in the bottom tool bar. Is this what you are seeing while in FF when the problem occurs?
What build of FF are you using, and are you having problems with any other browser while Avast is enabled?
Are you just surfing in FF or doing specific things like trying to DL when this happens? Do you have any software, like a firewall, where you are using FF in a Safe Mode that might be causing an issue?