The exterior of my home is made of dark brown brick. There are white shutters on the windows. At some time in the past, someone tried to either apply white paint or whitewash to the shutters. Below the shutters, on the brick, it looks as if white paint or whitewash has run down onto the brick. Does anyone have any idea what I can use to get rid of the run stains?

I have the same problem. I used muriatic acid and a stiff bristled brush. You can get it at Home Depot. It didn't completely remove stains, but faded them quite a bit.

Try paste paint remover and then hose off while wire brushing.