Seam-collapse damage occurs in the sealing layers of underground compressed air energy storage (CAES) caverns subjected to cyclic high internal pressure. To address this challenge, a system for evaluating flexible sealing materials was developed. Seam-collapse tests were conducted on butyl rubber under cyclic pressure to systematically investigate the evolution of damage and air permeability. After testing, surface damage was observed, followed by permeability tests. With increasing pressurization cycles, surface damage intensified, characterized by deepening depressions in the indentation area and the initiation of macroscopic cracking. In addition, the permeability coefficient increases during cycling, severely affecting the structural integrity and sealing function of the material. Increasing the material thickness effectively inhibits damage accumulation, thereby reducing the increase in the permeability coefficient during cycling. A strong correlation was observed between damage evolution and the permeability coefficient of butyl rubber, which was approximately described by a power-law model. By establishing relationships between the number of pressurization cycles and the permeability coefficient for different material thicknesses and by fitting the results to practical operating conditions, a relationship between sealing material thickness and fatigue life was obtained. A butyl rubber thickness of at least 6 mm is necessary to satisfy the sealing performance requirements for a CAES cavern with an internal pressure of 10 MPa and a designed minimum operational life of 30 years.