The Rishonim (early authorities) discuss whether a garment contain Shatnez (prohibited blend of wool and linen) is Mukse on Shabbat. According to MaHaram M'Rotenberg, it is Mukse, citing a proof from the Gemara in Masechet Besa. He argues that since it may not be worn neither on Shabbat nor during the week, it has no permitted function and is Mukse. The Aruch Hashulhan (Rav Yechiel Michel Epstein of Nevarduk, 1829-1908) clarifies that even according to this strict opinion, it is only classified as a "Keli She'm'lachto L'isur," and may be moved to make room or for other permitted functions. On the other hand, the Or Zarua cites the opinion that is not Mukse, since it is still classified as a vessel and has other functions it may serve, such as covering things.
Maran, in Siman 308, cites both opinions as a "Yesh V'Yesh" (Some say… and some say…"). In such a case the Halacha is in accordance with the later citation, which in this case is the lenient position. This is how the Menuhat Ahaba (Rabbi Moshe Halevy, Israel, 1961-2001) understands Maran.
Accordingly, if a person has a Shatnez suit hanging in his closet, it is permitted to move it to gain access to something else, according to both opinions. According to Maran it is not Mukse at all, and even according to the strict opinion, it is only Keli She'm'lachto L'isur which may be moved "L'sorech M'komo."
SUMMARY
A garment containing Shatnez is not Mukse, according to the letter of the law.
