The case is closed now.
The case is closed by now.
The case has been closed by now.
What exactly is the difference between them?
The case is closed now.
The case is closed by now.
The case has been closed by now.
What exactly is the difference between them?
Only the first sentence makes grammatical sense.
You would use 'by now' if you are talking about a hypothetical statement.
So for instance
The case 'should' have been closed by now.
In each of your three examples, the case has been closed, so you would just use 'now'.