An experimental investigation was conducted to determine the form and characteristics of damage that temporary fasteners may cause to the composite hole in the aviation temporary fastening process. According to the shapes of the temporary fastener clamping feet, copying indenters were designed. The similarity between the quasi-static indentation (QSI) experiment and the installation of temporary fasteners was discussed. The QSI experiment was used to investigate the influence of shapes of clamping feet, prepreg category, installation direction, and other factors on the failure of the composite hole, and the displacement-load curves were obtained. The damage morphology was observed with the optical microscope and ultrasonic microscope. The results show that the QSI experiment can simulate the temporary fastening damage well. Due to the expansion of delamination, and the final occurrence of fiber breakage in the composite hole, the bearing capacity is reduced. The temporary fastening damage process can be divided into three stages: in the elastic stage, the laminate does not suffer damage, and the surface resin appears slight dents; in the delamination stage, the layers delaminate at the edge of the contact surface, and expand in the direction of fiber; in the failure stage, sudden damage occurs suddenly under the indenter, and the fiber matrix is crushed into powder. The propagation range of the damage depends on the material properties and installation direction. Compared with T700/3234, T800/X850 has greater energy absorption before the damage. Laminates have the worst anti-intrusion ability out-of-plane in the surface fiber direction. These results increase the understanding of the out-of-plane compression damage of the composite material and provide a theoretical basis and method guidance for the control of the temporary fastening damage of the composite structures. The copying indenter designed in this paper can be used for further research on temporary fastening damage.