Fullerene molecules are cage-like nanoscopic structures with pentagonal and hexagonal faces. In practical applications such as fullerene-reinforced nanocomposites (FRNCs), these structures may be subjected to tension force. In this research, we employ molecular dynamics (MD) simulation to compute the behavior and deformation of different fullerene molecules, ranging from C60 to C2000, under tension force. To model the interactions between carbon atoms in the MD simulations, the adaptive intermolecular reactive bond order (AIREBO) force field is used.The displacement-force and the displacement-strain energy curves are obtained. It is observed that a new type of structural instability occurs in the fullerene molecules when the applied tension force increases. This abnormal structural instability in the fullerenes is investigated for the first time in the literature. The critical tensile forces and the corresponding mode shapes are determined for different fullerenes. The results indicate that the critical forces and deformations strongly depend upon the number of carbon atoms.