Update: with some serious help from Arunoda, I've learned that it was due to the react-google-ad component I was using bundling it's own copy of react. So, while the rest of the site was using production-mode React, that one component wasn't and that was enough to trigger the warning in the Chrome extension.
I sent a pull request to fix it: https://github.com/lukehansell/react-google-ad/pull/5