Bring 3 quarts water to a boil in a large saucepan. Add the beans. Cook until tender enough to be split with a fork, 8 to 10 minutes (cooking time will vary depending on the age of the beans). Drain. Set aside.
Pour the oil into a 10- to 12-inch skillet over medium heat. Add the onion. Cook until translucent, about 5 minutes. Stir in the fresh mushrooms and cook for 2 to 3 minutes more. Set aside.
Place the dried mushrooms in a 2-cup glass measure with ½ cup water. Cover tightly with plastic wrap and microwave on high for 3 minutes. Drain. With scissors, cut the dried mushrooms into ¼-inch pieces. Add with the snap peas to the onion-mushroom mixture and cook over medium heat for 3 to 4 minutes, stirring occasionally. When the snap peas are almost cooked through, add the cooked string beans. Season to taste with salt and pepper. Toss with the pasta.