Transport vehicles move with fluid drag, and the friction drag caused by turbulent flow is difficult to reduce. Therefore, many control strategies have been studied to reduce friction drag. The opposition control technique, which is one of feedback control techniques, has been shown to decrease the skin-friction drag of the turbulent flows by decreasing the activity of the turbulent vortical structures. However, the time delay from the detection of the velocity to the actuation should be compensated in the real word situation. Recently, nonlinear time series analysis based on the deterministic chaos property can be applied to wall turbulent flow for forecasting. In our group, we have applied the opposition control with the nonlinear forecasting for the turbulent channel flow to compensate the time delay, and a reduction in skin-friction drag has been achieved. In this study, we apply the opposition control with the nonlinear forecasting for turbulent Taylor-Couette flow using direct numerical simulations. In contrast to the previous studies, we focus on turbulent Taylor-Couette flow where the Taylor vortex appears. As the nonlinear forecasting, the orbital-instability-based forecasting method is employed. The flow is incompressible, the Reynolds number is set to 12000, and the cylindrical ratio is set to 0.5. Due to the opposition control with the nonlinear forecasting, the torque is reduced by 19.3% from the uncontrolled flow, while a pair of the Taylor vortex remains. The correlation coefficient between forecasted and actual velocities is larger than 0.8 in the region near the wall for not only the uncontrolled flow but also the controlled flow.