This paper presents a deep learning-based boundary element framework, termed DeepBEM, for the efficient and accurate numerical simulation of three-dimensional (3D) nonlinear heat conduction problems. The proposed framework is motivated by the limitations of conventional numerical approaches for nonlinear partial differential equations (PDEs), including numerical instability, high computational cost, insufficient convergence under strong nonlinearity, and sensitivity to initial guesses. The proposed method integrates the rigorous mathematical foundation of the classical boundary element method (BEM) with the expressive capability of deep neural networks. Instead of solving the resulting dense nonlinear systems as in conventional BEM, a neural network is employed to approximate the nodal temperature field, while the corresponding heat fluxes are obtained implicitly via automatic differentiation. The framework is formulated based on the boundary-domain integral representation of nonlinear heat conduction, which preserves the underlying physical laws and requires discretization primarily on the boundary. Domain integrals involving temperature-dependent nonlinear terms are evaluated using a moderate number of tetrahedral elements. The loss function is constructed exclusively from the residuals of the boundary integral equations (BIEs), thereby avoiding artificial weighting between governing equations and boundary conditions. Compared with conventional domain-based deep learning methods, the proposed method circumvents the explicit computation of high-order spatial derivatives and exhibits improved training stability, particularly for strongly nonlinear and multiphysics-coupled problems. Numerical examples of representative 3D nonlinear heat conduction problems demonstrate that the proposed method achieves accuracy comparable to high-fidelity reference solutions with robust convergence and significantly reduced degrees of freedom, highlighting its potential as an efficient computational tool for complex nonlinear thermal analyses in engineering applications.