Place potatoes in a medium saucepan with enough water to cover; bring to a boil over medium-high heat. Immediately reduce heat to a simmer; cook until potatoes are easily pierced with the tines of a fork, about 15 minutes. Drain, cool, and cut into quarters.
Meanwhile, bring a medium saucepan of salted water to a boil over high heat. Prepare an ice-water bath; set aside.
Add green beans to boiling water and cook until tender, about 4 minutes. Immediately transfer to ice-water bath to cool; drain.
Place potatoes, green beans, olives, celery, egg, dill, and tuna in a large bowl. Add vinaigrette and toss to combine; season with salt and pepper.
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