Not a teacher, nor a native.
I would say either he has had enough of his job and he might well be going to resign.
Or even he has already resigned.
As usual, I would strongly recommend you to wait until getting an answer from a teacher or a native.
Does "I'm done with sth" mean "I'm really tired of something" or "I have no more concern or interest in something"?
In the example you give, yes. But in other contexts it doesn't necessarily have such a dismissive tone, it can just mean "I've finished using": "You certainly can borrow my hammer, but I'm fixing the gate so you can have it when I'm done with it".