This seems to be related to the "Simplified Gear Lever" option in the "SPECIAL" options tab:
With "Simplified Gear Lever" checked (on), the aircraft starts with the gear lever in the UP position when spawning on the ground. This happens for both hot and cold starts. This can easily be verified using the "Cold Start" and "Take Off" instant action missions. In this configuration, pressing the "Landing Gear Up/Down" keybind after takeoff doesn't cause the gear to be retracted.
With "Simplified Gear Lever" off, the aircraft starts with the gear lever in the DOWN position, and the "Landing Gear Up/Down" keybind works normally (assuming the U/C safety lever is open).