How do I remove a contact from a group using the API?
From using the API Explorer (Support menu > Developer > API Explorer) I can see that you can delete a GroupContact entity by specifying a GroupContact ID and a Group Contact Status of "Removed".

I don't think that makes a lot of sense - even for a non-standard API such as GroupContact, since a Group Contact Status of "Removed" is NOT the same as a deleted contact.

Note that my findings are at variance with CRM-16878, which covers the same topic.