What is the closing date/ When is the closing date

Hi, Kipling. Welcome to the Forum!

"What is the closing date?" is correct.
The form with "when" is a less precise usage, but a lot of people also say it that way.

A similar case is when people say "Prices are getting so expensive!" Actually, more precisely, prices are getting higher, and products are getting more expensive.

Another one: The temperature is just too hot today!