For now, there is at least one workaround, but it only applies if you're using the Firefox Developer or Nightly build. If you look under "about:config" (if you don't know what this is, you probably shouldn't try it) and set the "xpinstall.signatures.required" value to False, then your extensions will start working again.

Update: If you're on another version of Firefox, there is a way to temporarily fix it, but you'll have to repeat it every time you reopen the browser. It involves enabling add-on debugging and manually loading each extension's .xpi file.

Update 2 (5/4, 4 AM ET): While the Mozilla team continues to work on resolving the issue, security researcher Kevin Beaumont mentions that anyone using the Tor browser will want to make sure they keep the NoScript extension enabled to browse securely. This issue has affected the installed-by-default extensions in that browser as well, and going without it could expose your information.

Update 3 (5/4, 7:30 AM ET): The team has announced a hotfix is rolling out now. It's applied in the background, so you shouldn't have to do anything to receive it. However, to make sure you get it right away, you can follow these instructions:

In order to be able to provide this fix on short notice, we are using the Studies system. You can check if you have studies enabled by going to Firefox Preferences -> Privacy & Security -> Allow Firefox to install and run studies. You can disable studies again after your add-ons have been re-enabled.