This research investigates the nonlinear pressure-induced stiffness characteristics of an origami structure associated to its large amplitude folding. Such structure consists of origami tubes that are formed by stacking and connecting Miura-Ori sheets along their crease lines, so that one can apply pressure to achieve autonomous folding and generate stiffness. We show that the stacked origami can exhibit two unique stiffness characteristics due to the nonlinear relationships between the internal pressure and folding deformation, as well as the interactions between pressurized tubes. One characteristic is negative stiffness, and the other is quasi-zero stiffness (QZS). Both of these stiffness characteristics are programmable by tailoring the Miura-Ori crease design, and controllable by adjusting the pressurization. This research would uncover the physical principles behind these stiffness characteristics, provide design guidelines, and discuss the potentials of developing the stacked origami into an adaptive structure with multiple and embedded dynamic functionalities.