This investigation addresses the dynamics of annular viscoelastic films flowing down a flexible tube. The fluid viscoelasticity is assumed to be weak in order to obtain approximate explicit expressions for the stresses. Based on Shkadov’s integral boundary layer method (Fluid Dyn., vol. 2(1), 1967, pp. 29–34), a set of nonlinear evolution equations is derived that is valid for flows with moderate Reynolds numbers. The linear stability property of the system is examined by using normal-mode analysis, which is verified by comparing the results with those resulting from the linearization of the full Navier–Stokes equations. The results indicate that the fluid viscoelasticity plays an unstable role in the stability of the annular film flow. The tube flexibility, which includes wall damping and wall tension, plays a dual role. A bifurcation analysis is performed, and the families of steady travelling waves are catalogued. It is found that the stiffness of the tube tends to stimulate the interfacial capillary ripples. The fluid viscoelasticity acts to strengthen the dispersion of the interfacial waves but weakens the interfacial capillary ripples. The spatio-temporal evolutions of the system are also solved numerically. When the tube radius is small enough, tube closure can be observed due to the Plateau–Rayleigh instability. The fluid viscoelasticity acts to promote tube closure while the tube radius is relatively small. However, it plays a role in postponing the closure of the tube with a large radius.