Progressive collapse leads to a decrease in the lifetime structural reliability. Numerical evaluation of the reliability of reinforced concrete (RC) buildings under progressive collapse, based on a three-dimensional model considering material and geometric nonlinear effects using a nonlinear dynamic analysis procedure, has not been conducted in recent studies. In this regard, a 5-story RC building is modeled nonlinearly, and progressive collapse is modeled as the instantaneous removal of the corner, external, and internal columns. Due to the uncertainty in the material properties and also structural damping, the sampling-based reliability analysis under the progressive collapse has been conducted. The plastic rotation of the beams adjacent to the removed column is considered as the engineering demand parameter. To determine the most important random variables in the reliability of the structure under progressive collapse, a reliability-based sensitivity analysis has been performed. Results indicated that the structural reliability as well as the importance of random variables depends on the position of the removed column and the expected performance level under progressive collapse. The highest reliability is obtained by corner column removal. For example, the reliability index corresponding to the performance level of collapse prevention in the case of corner, external, and internal column removal is 3.48, 2.96, and 3.17, respectively. In all situations, the structure was not able to provide the performance level of immediately occupancy. Sensitivity analysis results indicated that structural damping, yield strength and ultimate strain of steel, and concrete cover depth are the most important random variables affecting the structural reliability.