Is this a rental unit with the heater in a cabinet? If it re-lights relatively easily and burns clean blue when you've got the cabinet open, then I'd lean toward it being a vent issue. It lights when you open the door because you give fresh air.

If it's near the furnace, there might be other vent issues where the furnace is drawing air and blowing out the pilot. Check the ducts and maybe the filters.

Also verify the thermocouple rod is adjusted to the correct position relative to the pilot light flame. If it's too far away it'll shut off the gas.