Twitter’s flavicon is not showing on the new Firefox Quantum Beta. I checked and it works on Firefox and Chrome.
Don’t think the issue has been reported yet.
In principle, this should have nothing to do with Developer Tools per se.
Do you see the favicon on the tab? I just checked and the favicon is there (not on the location bar).
In these cases the best thing to do is to file a bug in Bugzilla. If it’s duplicated, it will be marked as such. If it’s not, someone will get to it! 
You can file it here: https://bugzilla.mozilla.org/enter_bug.cgi?product=Firefox&component=Tabbed%20Browser