This paper is concerned with a priori error estimates of a finite element method for numerical reconstruction of some unknown distributed flux in an inverse heat conduction problem. More precisely, some unknown distributed Neumann data are to be recovered on the interior inaccessible boundary using Dirichlet measurement data on the outer accessible boundary. The main contribution in this work is to establish the a priori error estimates in terms of the mesh size in the domain and on the accessible/inaccessible boundaries, respectively, for both the temperature $u$ and the adjoint state $p$ under the lowest regularity assumption. It is revealed that the lower bounds of the convergence rates depend on the geometry of the domain. These a priori error estimates are of immense interest by themselves and pave the way for proving the convergence analysis of adaptive techniques applied to a general class of inverse heat conduction problems. Numerical experiments are presented to verify our theoretical prediction.