"Set featured image" not working after 4.0 upgrade due to plugin

I’ve recently upgraded our multisite install, incrementally, from 3.2.4 to 4.0. All works fine except one thing: users cannot set featured images on their posts anymore. They can start the upload process, but are met by an error message saying “An error occurred in the upload. Please try again later.”

I’ve disabled all plugins and started enabling them one by one, and found that a custom plugin we use is the culprit. This plugin adds a new settings page for each site, with some text fields and checkboxes.

So, now that I know what is causing the problem, how do I figure out what to change in the plugin so it works properly? The plugin consists of a short .js file and about 500 lines of PHP. I can paste the code here if needed.