Drain chickpeas in a colander over a bowl, reserving 1 tablespoon liquid. Combine chickpeas, tahini, garlic, and red pepper in a food processor, and process until smooth, scraping sides of processor bowl once. Add lemon juice and reserved chickpea liquid, process until smooth. Spoon mixture into a bowl; set aside.
Combine tomato and next 6 ingredients (tomato through mint) in a bowl. Spoon 1/4 cup hummus mixture into each pita half; top with 1/4 cup feta-olive salsa.