I think I vaguely remember what the counterexpample was, but not the details. So if someone can fill it in it'd be great! (I'm putting this in CW mode for that reason.)

The idea is based on knowing explicitly the Green's function in the disk. The goal is to construct a continuous function $g$ on the boundary of the unit disk, such that it is the trace of some smooth function $f$ where $\triangle f = 0$ in the disk, and where the energy integral $\int_D |\partial f|^2 dx = \infty$.

The construction itself, I think, was based on finding a sequence of harmonic functions $f_k$ such that in the interior of the disk $\sum \partial^\alpha f_k$ converges pointwise for any derivative with multiindex $\alpha$ (but of course the sum fails to converge for any derivative on the boundary, so the final function only extends continuously to the boundary). the Trick is that these functions are chosen with specific boundary values, so that using the Green's function one can estimate (from below) the size of their gradients near the boundary. Then one just need to make sure that the blow-up rate of the gradient dominates the distance to the boundary, so the energy integral will fail to converge in a neighborhood of the boundary.

Since the energy integral of the solution to the Euler-Lagrange equation is infinite, the solution obviously is not a minimizer of the energy integral, thus contradicting Dirichlet's principle.