We study a realistic Floquet topological superconductor, a periodically driven nanowire proximitized to an equilibrium s-wave superconductor. Because of the strong energy and density fluctuations caused by the superconducting proximity effect, the Floquet Majorana wire becomes dissipative. We show that the Floquet band structure is still preserved in this dissipative system. In particular, we find that the Floquet Majorana zero and π modes can no longer be simply described by the Floquet topological band theory. We also propose an effective model to simplify the calculation of the lifetime of these Floquet Majoranas and find that the lifetime can be engineered by the external driving field.