It's strange, when I Google for the past 12 months there are no results for that exact phrase with Firefox:

https://www.google.com/search?tbs=qdr%3Ay&q=firefox+"Failed+to+load+the+Facebook+SDK"

Can you rule out Tracking Protection as an issue? Tracking Protection is applied by default in private windows. When Firefox blocks tracking sites, perhaps including the FB connect script, you should see a shield icon to the left of the address. This article has more info on how the feature works and the steps to make an exception for a site: What happened to Tracking Protection?.

No private window is used. Have a look at http://stackoverflow.com/questions/33855626/firefox-blocking-facebook-js for an explanation. I changed default.trackingprotection.enabled to false in about:config, reloaded the Twitter-page and it did the trick. And in Options - Privacy the "Use Tracking protection in Private Windows" checkbox is still checked. This solved my problem. And... the are lots of issues known if you do not use the exact search phrase. Also with Chrome it seems to be a well known problem.

Rick26 said

I changed default.trackingprotection.enabled to false in about:config, reloaded the Twitter-page and it did the trick.

I assume you mean

privacy.trackingprotection.enabled

which is "false" by default. I don't know how that got turned on in your Firefox.

If you want to keep tracking protection on for most sites, you could try using the shield icon to turn it off just for Twitter and other sites where you share your FB profile as your login method.