Place the green beans in a large saucepan with a lid or a slow cooker. Pour water over the beans to cover; stir in salt. Bring to a boil over medium heat, reduce heat to low, cover, and simmer until the beans are very tender, 3 to 4 hours; add water throughout the cooking time as needed to keep moist.
Add the smoked sausage and potatoes; cover and simmer until the potatoes are very tender and the beans and potatoes have absorbed the smoky flavor of the sausage, about 2 more hours.
