I'm in the same boat with Greenshot's Imgur Plugin. Greenshot is awesome.
According to Imgur Documentation
Status Code: 403
Forbidden. You don't have access to this action. If you're getting this error, check that you haven't run out of API credits or make sure you're sending the OAuth headers correctly and have valid tokens/secrets.
I assume in my case that many other people love Greenshot and the application just ran out of API credits or something? In your case it sounds like a broken authentication token though I'm knowledge about enough to know how to hypothetically even approach fixing it.
ShareX a similar screen shot program seems to have run into a similar problem at some point and the problem on their end was users had to reauthenticate. I'm going to try it on my end and if that works that it might be linked to the problem of hotlink 403 errors.
I wasn't able to relogin to Imgur but fiddling with the configuration settings (or waiting just 20 more minutes while I tried to figure out how to logout) seems to have fixed the issue.