Firefox doesn't ask me for a certificate when visiting a site that needs one

I have just been experimenting with this myself. It seems that Firefox will only prompt if you have already imported a certificate signed by the same issuer who has signed site's SSL certificate.

So get your CA to sign both the server and client certificates, then import the client certificate into Firefox (via the Preferences window.) The next time you visit the site, Firefox will notice that it has a client certificate signed by the same CA as the server's SSL certificate, so it will prompt you whether to use this or not.

I encountered this problem with Firefox, where it would only offer the certificate in a Private Browsing window.

I suspect this happened because I initially clicked Cancel on the User Identification Request dialog, with the Remember this decision box checked, as in this screenshot.

(Don't do this.)

I fixed this by going to Clear Recent History (CtrlShiftDel) and clearing the Site Preferences.

You might also be able to go to about:preferences#privacy -> View Certificates... -> Authentication Decisions, and Delete it from there: