A new method for the approximation of dissipative Euler-Maxwell equations is presented. These equations describe the interaction between a charged fluid and the surrounding electromagnetic field and it is well-known that they can be derived in the dissipation-less regime from Hamilton's principle. Following the work in [Gay-Balmaz&Yoshimura, 2017], irreversible processes can be included in an extended Lagrange--d'Alembert principle, leading to a formulation of viscous Euler-Maxwell equations with heat. The structure-preserving finite element numerical method is derived similarly by writing a discrete version of the continuous variational principle. It extends the approach in [Gawlik&Gay-Balmaz, 2024] for the Navier--Stokes-Fourier equations. The resulting numerical scheme satisfies a discrete version of the two laws of thermodynamics; namely, total energy is exactly conserved while the entropy is guaranteed to not decrease over time. Moreover, numerical dissipation is tightly controlled as the only entropy sources are physical, and entropy is exactly conserved when viscosity and heat are removed. The different properties of the scheme are confirmed on several test cases.