I am not positive why, but the AMO release did not work with Firefox Nightly.
However, after I installed https://addons.mozilla.org/en-US/firefox/addon/chrome-store-foxified/ and installed the version from the chrome webstore it worked without issue.
Thought I would share incase anyone else has issues or if there is something else going on.

It works fine for me in both v49 and v55.0b3 — both the current master and AMO.
The console errors in my console are GitHub's own.
Perhaps try in a new profile, there might be conflicts.
First thing I did was a new profile:

looks like it might be page specific.
The news feed as you show does work on mine as well. But the repository pages are not working
I have gotten a bit further, when visiting this issue page and having the debugger
"stop on exception" it stops with an SyntaxError Exception on
unreadIndicatorIcon@moz-extension://66e81471-ae34-184e-8fcc-4697b3c8ef52/mark-unread.js:237:6
which when looking at the source code looks quite a bit different
function unreadIndicatorIcon() {
const $notificationIndicator = $('.header-nav-link.notification-indicator');
const $notificationStatus = $notificationIndicator.find('.mail-status');
let hasNotifications = $notificationStatus.hasClass('unread');
/* FOLLOWING IS LINE 237 (according to ff) */
if (JSON.parse(localStorage.unreadNotifications).length > 0) {
hasNotifications = true;
$notificationStatus.addClass('local-unread');
} else {
$notificationStatus.removeClass('local-unread');
}
$notificationIndicator.attr('aria-label', hasNotifications ? 'You have unread notifications' : 'You have no unread notifications');
}
which does seem like a syntax error...
What are you using exactly? That code is not there anymore:
function unreadIndicatorIcon() {
const notificationIndicator = select('.header-nav-link.notification-indicator');
const notificationStatus = notificationIndicator.querySelector('.mail-status');
const hasUnread = select.exists('li.js-notification.unread') || loadNotifications().length > 0;
const label = hasUnread ? 'You have unread notifications' : 'You have no unread notifications';
notificationStatus.classList.toggle('unread', hasUnread);
notificationIndicator.setAttribute('aria-label', label);
}
https://github.com/sindresorhus/refined-github/blob/f073dd2/src/libs/mark-unread.js#L236-L244
Ah. 1.20.1 is not on AMO. Investigating...
The extension's ID did not match the ID on AMO, so AMO thought it was an unlisted extension and it just signed it:

@sindresorhus I manually submitted 1.20.1
1.20.1 is now on AMO. Update!
@bfred-it I knew the code wasnt there that is why I said it didnt look the same :)
I can confirm this is now working thank you
Excellent! Thanks :)
Most helpful comment
1.20.1 is now on AMO. Update!