We investigate magnetic excitations in iridium insulators with pentavalent Ir5+ (5d4) ions with strong spin-orbit coupling. We obtain a microscopic model based on the local Ir5+ multiplets involving J=0 (singlet), J=1 (triplet), and J=2 (quintet) spin-orbital states. We get effective interactions between these multiplets on square and face-centered-cubic (fcc) structures of magnetic ions in the layered-perovskites and the double-perovskites, in particular Ba2YIrO6. Further, we derive an effective spin-orbital Hamiltonian in terms of bond bosons, and we explore possible instabilities toward magnetic and quadrupole orderings. Additionally, we study charge excitations with the help of the variational cluster perturbation theory, and we calculate the electronic charge gap as a function of hopping and Coulomb interactions. Based on both electronic and magnetic phase diagrams, we verify the possibility of excitonic magnetism due to condensation of spin-orbital excitons in Ir5+ iridates.