Precise control of tensile stress and intrinsic damping is crucial for the optimal design of nanomechanical systems for sensor applications and quantum optomechanics in particular. In this letter, we study the influence of oxygen plasma on the tensile stress and intrinsic damping of nanomechanical silicon nitride resonators. Oxygen plasma treatments are common steps in micro and nanofabrication. We show that oxygen plasma for only a few minutes oxidizes the silicon nitride surface, creating several nanometer thick silicon dioxide layers with a compressive stress of 1.30(16) GPa. Such oxide layers can cause a reduction in the effective tensile stress of a 50 nm thick stoichiometric silicon nitride membrane by almost 50%. Additionally, intrinsic damping linearly increases with the silicon dioxide film thickness. An oxide layer of 1.5 nm grown in just 10 s in a 50 W oxygen plasma almost doubled the intrinsic damping. The oxide surface layer can be efficiently removed in buffered hydrofluoric acid.