Make sour mix: Combine lime juice, lemon juice, and simple syrup. Set aside.
Make tangerine simple syrup: Strongly steep tea bags in boiling water. Discard bags and add sugar. Simmer, stirring, until sugar dissolves. Cool.
Add ice to a cocktail shaker and add bourbon, 2 tablespoons sour mix, and 2 tablespoons tangerine simple syrup. Shake well. Strain into a tumbler over ice. Garnish with cherry and orange.