What would be an instance where we can only use "What do you care?". Can you please provide one example if you can think of it otherwise it's okay as I've already got your point.
It seems to me that if the speaker genuinely wants to know why someone cares about something then "Why do you care?" is the only choice. "What do you care?" is normally a sarcastic or dismissive remark, and the speaker is not really interested in any answer.