That can happen when Firefox’s “Tracking Protection” is enabled — it blocks major social buttons, and is off by default in Firefox’s regular browsing mode. You can toggle it as outlined here:
https://support.mozilla.org/en-US/kb/tracking-protection
If your site needs share buttons when that option is enabled, you can use the “Cache AddToAny locally with daily cache updates” options in Settings > AddToAny > Advanced Options.