Poroelasticity inspects the interaction among fluid flow and elastic deformations within a porous medium. In several applications in the context of environmental sustainability, such as geothermal energy production and CO2 sequestration, temperature plays a key role in the description of the physical phenomena. Thus, in order to correctly describe these geological processes, the differential problem should also take into account the influence of the temperature, leading to a fully-coupled thermo-poroelastic (TPE) system of equations. We present and analyze a discontinuous Galerkin method for the numerical modelling of the non-linear fully-coupled (quasi-static) thermo-poroelastic problem. For the spatial discretization, we design a high-order symmetric weighted interior penalty scheme that supports general polytopal grids and is robust with respect to strong heteorgeneities in the model coefficients. We focus on the treatment of the non-linear convective transport term in the energy conservation equation and we propose suitable stabilization techniques that make the scheme robust for advection-dominated regimes. The stability analysis of the problem and the convergence of the fixed-point linearization strategy are addressed theoretically under mild requirements on the problem's data. A complete set of numerical simulations is presented in order to validate the theoretical analysis and to inspect numerically the robustness properties of the proposed method.