Where does it appear, exactly? One possibility is that you have a Pin It "bookmarklet" which is a script saved onto the Bookmarks Toolbar. If you right-click the button, does it have a menu with options like:

Open Open in a New Tab ... New Bookmark... ... Delete

If so, use Delete to remove it.

If it doesn't have any of those, hmm, the plot thickens.

I'm afraid the plot thickens. Before this started Ant Video Downloader had just updated, but before I posted this and provided info I had tried removing it. I also have AdBlock Plus, but the thought of that carrying anything is... as troubling as it is unlikely.

Does the button actually work -- it's not just a leftover from something?

You can move a button off a regular toolbar by right-clicking it or dragging it to the unused icons on the Customize tab, but if it comes back, that suggests there's active add-on code somewhere. See: Customize Firefox controls, buttons and toolbars.

Or do you want to post a screenshot of the button? This article has tips on capturing screenshots: How do I create a screenshot of my problem? Please cut out or blur any sensitive information in the image before attaching it to your reply.

Oh! Oh dangit, I am an idiot. It's not the browser, it's the website that I frequent. They've done something to partner with Pinterest and it took me until now to notice. The "Pin It" button only appears upon their website, and since it's a website element, I have no control over it.

To answer your question, however, yes the button works. Also, thank you for trying to help, but I'm afraid I was just un-observant-ly dumb.

Phew, I was running out of ideas.

There are add-ons directed toward blocking social/sharing buttons in websites because they can be used for tracking. But whether that would work on this particular site, I don't know.