Before revision:  We prove that the k-List Coloring, List k-Coloring, and k-Precoloring Extension problems are NP-complete on bipartite graphs with diameter at most d, for every k 4 and every d 3, and for k=3 and d 4, and that List k-Coloring is polynomial when d=2 (i.e., on complete bipartite graphs) for every k \geq 3. Since k-List Coloring was already known to be NP-complete on complete bipartite graphs, and polynomial for k=2 on general graphs, the only remaining open problems are List 3-Coloring and 3-Precoloring Extension  when d=3.
After revision:  We prove that the k-List Coloring, List k-Coloring, and k-Precoloring Extension problems are NP-complete on bipartite graphs with diameter at most d, for every k 4 and every d 3, and for k=3 and d proving NP-completeness in most cases, and leaving open only the List 3-Coloring and 3-Precoloring Extension when d=3.