Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs) with closed control regions are formulated and studied. A new duality principle between the linear backward stochastic Volterra integral equations and a class of linear stochastic integral equations with nonadapted solutions is derived, which extends and improves the corresponding results in [Y. Shi, T. Wang, and J. Yong, Math. Control Relat. Fields, 5 (2015), pp. 613{649], [J. Yong, Probab. Theory Related Fields, 142 (2008), pp. 21{77]. Some first order necessary optimality conditions for optimal controls of FBSVIEs are established via these duality principles. Instead of using the spike variation method as one may imagine, here we choose to treat the nonconvexity of the control regions by borrowing some tools in set-valued analysis and adapting them into our framework. In contrast with existing routines to deal with the nonconvexity in stochastic control problems, here only one adjoint system and one-order differentiability requirements of the coefficients are needed.