Approximate analytical bound-state solutions of the Dirac particle in the fields of attractive and repulsive Rosen-Morse (RM) potentials including the Coulomb-like tensor (CLT) potential are obtained for arbitrary spin-orbit quantum number kappa. The Pekeris approximation is used to deal with the spin-orbit coupling terms kappa (kappa +/- 1)r(-2). In the presence of exact spin and pseudospin (p-spin) symmetries, the energy eigenvalues and the corresponding normalized two-component wave functions are found by using the parametric generalization of the Nikiforov-Uvarov (NU) method. The numerical results show that the CLT interaction removes degeneracies between the spin and p-spin state doublets.