Preheat oven to 425°F. Wrap beets in foil and bake until tender when pierced, about 1 hour. Set aside to cool. Peel beets and cut into small wedges.
Bring medium saucepan of lightly salted water to a boil. Add haricots vert and cook until tender, about 4 minutes. Using slotted spoon, transfer haricots vert to colander and rinse under cold running water; pat dry with paper towels. Pour off all but 3 inches water from pan and place steamer basket inside. Add potatoes, cover and steam over medium-high heat until tender when tested with a fork, about 20 minutes. Let cool slightly, then cut each potato into quarters.
Make dressing: In medium bowl, combine shallots, horseradish, vinegar, sour cream and mustard. Whisk in oil, 1 tablespoon at a time, until well blended. Season with salt and pepper. (Vegetables and dressing can be prepared up to 1 day ahead. Cover separately and refrigerate. Bring to room temperature before using.)
Just before serving, stir dill into dressing. In large bowl, toss beets, beans and potatoes with 1/4 cup of dressing. Mound beet salad on plates and garnish with eggs, if using, and tomatoes. Serve remaining dressing separately.