Bring a medium pot of well-salted water to a boil. Add potatoes and cook until tender but not falling apart, about 10 minutes. Just before potatoes are done, add green beans to the pot for 30 seconds. Drain and set aside.
Whisk mustard, vinegar, 2 teaspoons oil, sugar, and 1/4 teaspoon salt in a bowl.
Heat remaining oil in a large skillet over medium-high heat. Add leek and dill, and saute, stirring occasionally, until golden and slightly crispy, 4 to 5 minutes.
Gently toss potatoes and green beans, celery, cucumber, tofu, chives, and half the leeks with the mustard dressing in a bowl. Adjust salt if necessary. Top with remaining leeks. Serve chilled or at room temperature.