A balanced-force control volume finite element method is presented for three-dimensional interfacial flows with surface tension on adaptive anisotropic unstructured meshes. A new balanced-force algorithm for the continuum surface tension model on unstructured meshes is proposed within an interface capturing framework based on the volume of fluid method, which ensures that the surface tension force and the resulting pressure gradient are exactly balanced. Two approaches are developed for accurate curvature approximation based on the volume fraction on unstructured meshes. The numerical framework also features an anisotropic adaptive mesh algorithm, which can modify unstructured meshes to better represent the underlying physics of interfacial problems and reduce computational effort without sacrificing accuracy. The numerical framework is validated with several benchmark problems for interface advection, surface tension test for equilibrium droplet, and dynamic fluid flow problems (fluid films, bubbles and droplets) in two and three dimensions.