Cu-based catalysts supported on sepiolite have been tested in vapor-phase hydrogenation of furfural. The incorporation of basic or amphoteric metal oxides (magnesium oxide, zinc oxide, or cerium oxide) improves the catalytic behavior, reaching a maximum furfural conversion above 80% after 5 h of reaction at 210 °C. In all cases, the main product is furfuryl alcohol, obtaining 2-methylfuran in lower proportions. The incorporation of these metal oxide species ameliorates the dispersion of metallic Cu nanoparticles, increasing the number of available Cu⁰-sites, which enhances the catalytic performance. The presence of acid sites favors the hydrogenolysis of furfuryl alcohol towards 2-methylfuran, although it also causes an increase of carbon species on its surface, which is associated with the catalytic deactivation of the catalyst along the time-on-stream.