Viscous bubbles are prevalent in both natural and industrial settings. Their rupture and collapse may be accompanied by features typically associated with elastic sheets, including the development of radial wrinkles. Previous investigators concluded that the film weight is responsible for both the film collapse and wrinkling instability. Conversely, we show here experimentally that gravity plays a negligible role: The same collapse and wrinkling arise independently of the bubble's orientation. We found that surface tension drives the collapse and initiates a dynamic buckling instability. Because the film weight is irrelevant, our results suggest that wrinkling may likewise accompany the breakup of relatively small-scale, curved viscous and viscoelastic films, including those in the respiratory tract responsible for aerosol production from exhalation events.