I believe this is a problem happening on facebook's end. But, there is a way you can fix it... Just copy the URL of your page and enter it in the Facebook LINTER tool here:
Facebook LINTER -> https://developers.facebook.com/tools/debug
I'm not sure of the exact details of why this works, but from what I've been able to gather, when you run the LINTER tool on your URL it will also update the cache facebook has for your URL. The error with the button working is on the facebook side, so by running this tool and getting facebook to update their cached data for your page, the like button starts working again. I've seen this broken facebook button occur for a few different reasons, one being that a page used to use the old facebook share button. When you add the new like button to a page that used to use a share button, the like button won't work until you run the page through the linter.
Bottom line is doing this will fix your like button. If your facebook button doesn't work on a page, run it through the LINTER tool. Hope this helps!