If I were bartending ,would it sound foreign to ask a customer "What would you be drinking today?" instead of "What are you drinking?"?
It would sound as if you were trying to put on some mock-Irish manner of speech.
'What are you drinking?' is more normal from somebody who is offering to buy you a drink, or is simply curious about the drink in front of you.
A bartender is more likely to ask, 'What would you like?', 'What can I get you?', 'What'll it be?' or just 'Yes, sir?'