Graphene is an intriguing material in view of its unique Dirac quasi-particles, and the manipulation of its electronic structure is important in material design and applications. Here, we theoretically investigate the electronic band structure of epitaxial graphene on SiC with intercalation of rare earth metal ions (e.g., Yb and Dy) using first-principles calculations. The intercalation can be used to control the coupling of the constituent components (buffer layer, graphene, and substrate), resulting in strong modification of the graphene band structure. It is demonstrated that the metal-intercalated epitaxial graphene has tunable band structures by controlling the energies of Dirac cones as well as the linear and quadratic band dispersion depending on the intercalation layer and density. Therefore, the metal intercalation is a viable method to manipulate the electronic band structure of the epitaxial graphene, which can enhance the functional utility and controllability of the material.