Can oil from a pine tree stain my exterior paint?
I bet what you are seeing is mildew growing on dust on the wall.  The dust can be formed of the white blush on the needles of the pine, or it may jsut be taht the pine protects the house from wind, so dust collects there.

I would expect dripping to be, well, more drip like -- more blobby.

If it's mildew, it should come off with a weak bleach solution.  (1 cup per gallon of water, along with a table spoon of TSP per gallon.  You may have to wash it once with the bleach, wait a day, then wash with the TSP.

it's actually sap, it is essentially oil based.  Try winter windshield washer fluid, about 40% alchohol, or diesel fuel, and a scotchbrite nylon scrubby.   DON'T use gasoline.