The Shulhan Aruch establishes in several places that it is permissible to touch Mukse items, as long as not even a part of it moves. For example, Maran rules that a corpse is Mukse, and even though a single limb may not be moved, it is permissible to touch it.
The Shulhan Aruch (306:6) records the Halacha that covering a Mukse item with a vessel is permitted, only if he does not touch the Mukse itself. The commentaries wonder what the problem is, since it is permitted to touch Mukse. The Maggid Mishne understands that the case is referring to a Mukse item which is round, and touching it will cause the item to roll on its axis. The Trumat Ha'deshen (Rav Yisrael Isserlin, 1390-1460) offers an alternate explanation: Touching Mukse is prohibited in a case where the purpose of the contact is to benefit the Mukse item. The case of placing a protective covering over the Mukse item is for the benefit of the Mukse item, and that is why it is prohibited.
Thus, there is a disagreement between the Maggid Mishne and the Trumat Ha'deshen whether one may touch a Mukse item for its benefit. The Bet Yosef brings both positions. The Magen Abraham (Rabbi Abraham Gombiner, Poland, 1637-1682) is stringent in accordance with the Trumat Ha'deshen. However, many Poskim rule in accordance with the Maggid Mishne, including the Vilna Gaon and the Mishna Berura.
Hacham Ovadia brings a proof to be lenient from the ruling of Maran who permits anointing a corpse with oil on Shabbat, even though the corpse is being touched for its benefit. This is also the opinion of Teshubot Bene Sion.
It is permissible to touch a Mukse item for its benefit, as long as no part of it will move.