Preheat a grill to medium high. Mix the butter, paprika, lemon zest, 2 pinches of salt and a grind of pepper in a small bowl until smooth; set aside. Toss the green beans, wax beans, butter beans and garlic with the olive oil, 1 teaspoon salt and a few grinds of pepper in a large bowl.
Lay out two 24-inch-long sheets of heavy-duty foil. Divide the vegetables between the foil sheets. Add 1 tablespoon water to each mound of vegetables. Bring the short ends of the foil together and fold to seal, then fold in the sides and seal tightly to make 2 packets.
Grill the foil packets, 8 minutes. Flip the packets and continue grilling until the beans are tender and charred in spots, about 4 more minutes. Carefully open the packets and toss with the paprika butter. Serve with lemon wedges.