Not only does Avast scan activity on your hard drive and web browsers, but it also monitors POP3 and IMAP email clients, including Apple Mail, Thunderbird, Postbox and Airmail.

I assume Safari is working normally on those sites?

Mac Mail And Gmail Duplicate Email

The latest version of avast! Web Shield includes a feature to filter your web connections for potential threats. In order to filter secure sites, it must present a 'fake' certificate for the site to Firefox, so it can be 'in the middle' and read your secure communications. If Firefox is not set up to trust the fake certificates, then you will get warnings for every secure site.

You might want to turn this feature off as a test to see whether the problem goes away.

If that works, but you want to use the feature, you'll need to import Avast's certificate into Firefox's 'Authorities' tab so Firefox accepts its fake certificates (there might be an automated way AVAST intended for you to do this, or you might need to export from IE and then import into Firefox).

Mac Mail And Office 365

Here is some discussion on avast!'s forum about how this is supposed to work: Issues and questions on HTTPS Scanning - Avast Internet 2015