Why do my Ford Crown Victoria headlamps turn off when engine warms up?

you have a bad ground somewhere. It's not that the engine is heating up its that electricity is flowing and heating up the poorly grounded area. More heat causes more resistance which will prevent your lights from working. Check the ground on your engine and where your headlights ground out.

Thank you, Patrick. I suspected the same, but have waited for confirmation.

There was a Ford recall of crown vics with this exact problem -- mine was one of them. Check with your dealership and they can tell you if this is the problem and will fix free!!