How fast does an airplane need to fly to chase the sunset but never catch it?

Technically, it depends on your location relative to the Earth, but a rough estimate would be 765 MPH at the Tropic of Cancer / Capricorn (18,380 miles / 24 hours). At the equator, it would be 1,037.57 MPH (24,901.55 miles / 24 hours). At the poles, it would be zero.