YouTube responded with an error: There is a per-IP or per-Referer restriction configured on your API key and the request does not match these restrictions. Please use the Google Developers Console to update your API key configuration if request from this IP or referer should be allowed.
We've seen this error in just a handful of cases. It's an error from Google and it's not very clear why it happens with some keys and not others. The permanent solution in every case so far was to redo the initial setup process to obtain a new API key. It's very important to follow those instructions precisely, as getting the wrong type of API key could also cause issues. Give that a try and let us know how it goes?
We understand that this setup process can be annoying, and I should note that we offer TubePress Express: a cloud-hosted version of TubePress that relieves you from this setup. TubePress Express has zero-installation, is maintenance free, and you can use it on as many of your websites as you like. Something to consider, perhaps.