Addressing the link you cited, Tokugawa Ieyasu taking no part in fighting is not the same as opposing the war in general. In fact, Ieyasu was the one who proposed the invasion strategy that Hideyoshi adopted. When combat operations began, Tokugawa troops were part of the reserves who stayed in Kyushu. But, as you said, whether or not Ieyasu actually opposed Hideyoshi's plan, he definitely ended up staying out of the fighting.
The immediate (and official) reason is because the Tokugawa clan is occupied. Keep in mind Ieyasu had only just been relocated to the newly vacated lands of the the Hojo clan. Hideyoshi's (ultimately failed) move uprooted the Tokugawa clan from their native land, and forced them to under the difficult process of integrating themselves into the Kanto provinces. Not only did Ieyasu now need to settle his retainers and their families, he also have to establish an administration in his new territories. Furthermore, he would have to pacify the provinces.
Moreover, not only did the Tokugawa clan face a great deal of work, having been generously rewarded already they also cannot expect to be rewarded with any more land. This contrasts with the western daimyos, who were looking forward to being rewarded with Korean conquests. Expending soldiers in that war thus would have been a waste for Tokugawa ambitions. Fortunately, because of Hideyoshi's ill-fated gift, Ieyasu could plead to be exempted from military service using the home front's difficulties as an excuse.
So while the western daimyos whittled their strength away on the fields of Korea for eight bloody years, the Tokugawa armies rested and managed their territoy of Kanto into the basis for their later victory over the Toyotomi clan.