Bring a large pot of salted water to a boil. Meanwhile, toss the zucchini with 1/2 teaspoon salt in a colander set over a large bowl. Let stand 10 minutes, then gently squeeze out the excess moisture.
Add the pasta to the boiling water and stir vigorously to prevent it from sticking. Cook until al dente, about 12 minutes. Reserve 1/2 cup of the cooking water, then drain the pasta.
Meanwhile, heat the butter in a large skillet over medium-high heat. Add the cherry tomatoes and cook until blistered and slightly softened, about 4 minutes. Stir in the zucchini and lemon zest and cook, lightly crushing the tomatoes with a wooden spoon, until the zucchini is crisp-tender, about 4 minutes. Season with salt and pepper.
Transfer the zucchini-tomato mixture to a large bowl. Add the pasta and cheese and toss. Stir in half of the chives and about 1/4 cup of the reserved cooking water, adding more to loosen, if needed. Season with salt and pepper. Divide among bowls and top with more cheese and the remaining chives.