We consider the Poisson model problem with inhomogeneous Dirichlet boundary conditions. For the numerical solution, we employ the boundary element method (BEM). Unlike existing work, we aim for a-posteriori error control of the potential error (in the domain) instead of the error of the approximated integral density (on the boundary). To this end, we employ the well-known technique of functional error estimates. One key feature is that the derived error estimates are independent of the BEM discretization and provide guaranteed lower and upper bounds for the unknown error. In particular, the analysis covers Galerkin BEM and the collocation method, what makes the approach of particular interest for scientific computations and engineering applications. Numerical experiments for the Laplace problem confirm the theoretical results.