Place tomato halves upside down on paper toweling to drain, about 10 minutes.
Meanwhile, cook pasta in large pot following package directions. After cooking 5 minutes, add green beans; continue cooking until pasta is al dente and beans are crisp-tender. Drain pasta and beans in colander; rinse under cool running water. Drain well. Transfer pasta and beans to large bowl. Add grilled chicken.
Stir together mayonnaise, pesto and vinegar in small bowl. Add to pasta mixture along with salt and pepper. Gently toss to combine.