When a gas phase with low viscosity displaces the fluid contained in a capillary, it forms a thin liquid film lining the inner wall. The resulted liquid film is unstable subject to certain axisymmetric perturbations, and eventually breaks up into regularly spaced lobes or liquid meniscus occluding the capillary, depending on its initial film thickness. This phenomena is further complicated by the existence of surfactants on the interface and the shear thinning behavior of lining liquids. Many analytical and numerical studies have been reported regarding the critical film thickness, larger than which meniscus occlusion forms, and the effect of surfactants, the existence of which significantly affects the growth rate of instability and thus the closure time. However, the shear-thinning effect of fluid was seldom considered despite the large volume of literature. In the current work, we studied the shear-thinning effect of lining fluid by simultaneously solving the coupled set of 2D axisymmetric continuity and momentum equations governing the fluid flow and the 1D convection-diffusion equation governing the transport of insoluble surfactant on the interface using the Galerkin finite element method. The shear-thinning behavior of fluid is described using a three-parameter Carreau model. We found that the critical film thickness and closure time are reduced for shear-thinning fluids when compared with Newtonian fluid with same surfactant elasticity number and initial surfactant concentration. Our results also show that the shear-thinning effect is less significant when the surfactant is weaker.