Which is grammatically correct 'on what date' or 'on which date'?

"On a date" is correct when talking about going out with someone. You would say "Tom is going on a date with Betty," not "Tom is going in a date with Betty."

"In a date" would be used when talking about the fruit date and something inside of the date. "There are seeds in the date," or "Molly put goat cheese in the date."