There are daytime running lights in the vehicle, the only way to get them to turn off, is to pull up on the emergency brake. If you pull the break up one click, then it will turn the lights off, and you might be able to still drive, but I'm not sure.
pull the headlight fuse
check the headlamp relay most likely stuck
if you have the owners manual it tells you how to change (turn off) the auto lights on setting basically turn the light control knob
Because 99 model has automatic headlights, in order to turn it off you have to turn the headlight switcher off, and also remove keys from the ignition.
On a 1999 Ford Windstar, the headlights might come on when the car is shut off because of a mechanism that allows for a courtesy light. The headlights come on for a few seconds to allow time for the driver to enter a building.
Check fuse for parking/tailights common circuit
this model line has problems with the brake light switch-- there was a recall--
The wires might be grounding out at night when you turn them on, just pull them out and check and also it might be a short.
Check the ground wire to the tail lights.
Your headlights might stay on even when the key is removed from the ignition and the switch is off because this is a safety feature in the 1999 Dodge Neon. You can use your vehicle's manual to find out how to turn off this feature if it bothers you. The lights should shut off after a few minutes.
If it's like my '99 model, it's a collar around the headlight pull-switch. Just turn the collar portion all the way counterclockwise until it clicks off. The headlights will then turn off anytime the car is in park.
replace the head light switch