We tracked it down! There are two issues:
- Wikimedia now commonly blocks action=login requests that uses regular passwords
- Wikimedia now blocks requests missing user agents
We do provide user agents for our edit requests but not for the login request which is causing Wikimedia to refuse all those requests since a few days ago.
The user-agent issue needs a fix on the OpenRefine side and for the password one we probably want do do some UI changes.