We study the $Q^2$ evolution of the chiral-odd twist-3 spin structure function, $h_L(x,Q^2)$, which is relevant for the nucleon-nucleon polarized Drell-Yan process, in the leading logarithmic approximation of the QCD perturbation theory. We pay particular attention to the operator mixing including the equation-of-motion operators to calculate the anomalous dimension matrix for the twist-3 operators. The result is compared with those for the twist-2 distributions and the other twist-3 structure function $g_2(x,Q^2)$.