We consider a magnetic insulator in contact with a normal metal. We derive a self-consistent Keldysh effective action for the magnon gas that contains the effects of magnon-magnon interactions and contact with the metal to lowest order. Self-consistent expressions for the dispersion relation, temperature, and chemical potential for magnons are derived. Based on this effective action, we study instabilities of the magnon gas that arise due to spin current flowing across the interface between the normal metal and the magnetic insulator. We find that the stability phase diagram is modified by an interference between magnon-magnon interactions and interfacial magnon-electron coupling. These effects persist at low temperatures and for thin magnetic insulators.