The Finite Element Tearing and Interconnecting (FETI) method is an iterative substructuring method using Lagrange multipliers to enforce the continuity of the finite element solution across the subdomain interface. Mortar finite elements are nonconforming finite elements that allow for a geometrically nonconforming decomposition of the computational domain into subregions and, at the same time, for the optimal coupling of different variational approximations in different subregions. We present a numerical study of FETI algorithms for elliptic self-adjoint equations discretized by mortar finite elements. Several preconditioners which have been successful for the case of conforming finite elements are considered. We compare the performance of our algorithms when applied to classical mortar elements and to a new family of biorthogonal mortar elements and discuss the differences between enforcing mortar conditions instead of continuity conditions for the case of matching nodes across the interface. Our experiments are carried out for both two and three dimensional problems, and include a study of the relative costs of applying different preconditioners for mortar elements.