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.
He stayed out because he was saving his strength for taking over Japan. Right after the Korean war finished he staged a coup and took control from Hideyoshi's government.
To Semaphore's excellent answer I would just add that Hideyoshi was already showing signs of declining health. Ieyasu's Japanese nickname can be translated as "cunning badger" and he is known for patience and long term outlook. Thus, it is likely that he envisioned ambitious Western daimyo returning with battle fatigued troops and a physically (and indirectly mentally) weakened Hideyoshi who at the time of the planning didn't have any natural heir, a definite disadvantage in clan conscious Japan. As it was, Hideyori was born as the invasion was under way, and this might have influenced both Hideyoshi and Ieyasu. The former now wanted to establish a dynasty, the latter planning moves to oppose it.
Ieyasu left reserve troops at Nagoya castle (the one on Kyushu) but spent most of his time in Edo, preparing for the future. It is no coincidence that once Hideyoshi died, he moved swiftly to establish himself as the hegemon of Japan.