Combine tomatoes, 1/3 cup chives, 1 tbsp garlic, vinegar, 3/4 tsp salt, and pepper in a large bowl; let stand for 15 minutes. Drain the mixture in a colander over a bowl, reserving liquid.
While the tomatoes stand, bring 2 quarts water to a boil in a large saucepan. Add 1 tablespoon salt and pasta. Cook the pasta for 10 minutes or until al dente, and drain.
Heat butter, olive oil, and remaining 1 1/2 tablespoons garlic in a large skillet over low heat; cook 4 minutes or until fragrant. Increase the heat to medium-high.
Add the reserved tomato liquid, and bring to a boil; cook until reduced to 1/2 cup (about 6 minutes). Add clams; cover and cook 4 minutes or until the shells open. Remove the clams from the pan, and discard any unopened shells.
Add the reserved tomato mixture and pasta to the pan; cook for 2 minutes or until thoroughly heated. Top with the remaining chives and clams.