This should be fixed in a future version of Firefox (timing is unknown at this time).

For now, I only know about the registry change option, described in this thread:

unable to open this internet shortcut. the protocol "FirefoxURL" does not have a registered program

It's possible to import a .reg file to update your registry, or to use a script or program to do it, but you would want to find a trusted source for that.

Thanks, jscher2000. I already knew about that thread, but it was Greek to me. What I find so frustrating is that it worked FINE two days ago before the update I did. And the shortcuts work fine on the desktop! That just blows my mind. Should I use "System Restore" to go back before I updated Firefox?

Hi Yellowcopter, it depends on how critical that Pinned sites list is to you.

If you can't live without it, then you could roll back to Firefox 23.0.1 (recognizing that there are some security vulnerabilities that Mozilla already disclosed).

If you can use desktop shortcuts or Firefox bookmarks to get by until it's fixed (hopefully in Firefox 25), then that would be preferable.

By the way, I don't recommend using system restore without making a backup of your personal data. There is a possibility of losing recent bookmarks or other data that you might miss. This article has tips on backing up: Back up and restore information in Firefox profiles.

Ok, I guess I'll just wait until the unseen powers at Mozilla fix this glitch that they've managed to introduce into their otherwise-great browser!