Cook potatoes in boiling water 30 minutes or until tender; drain and cool 10 minutes. Cut potatoes into 1-inch chunks and place in a shallow dish. Cook beans in boiling water 10 minutes or until tender; drain and cool 10 minutes. Add to potatoes. Combine oil, vinegar, chives, salt and pepper in a small jar with tight-fitting lid. Shake well and pour two thirds of the dressing over potatoes, and beans. Refrigerate, covered, 2 hours or up to 24 hours, stirring occasionally.
To serve, arrange lettuce on four plates and spoon potato/string bean mixture on top. Divide crabmeat among the plates and garnish with olives and tomatoes. Drizzle remaining dressing on top.