A native speaker would say: 'What's the date today?' or just 'What's the date?'

Just as the most natural-sounding English sentence using your set of sentence options: "Tell me the date" is correct. However, I agree with Paul: if you're asking for today's date "What's the date?" or "What's the date today?" is preferred. If you're asking about a specific event, it's probably more common to just ask "When is it?" Example: "You should come to the staff party." "I'd love to. When is it?"

"Tell me the date" is the best, I think. :)