Rabbi Sorotzkin's Insights in Torah writes about the meil (coat or jacket) that the Kohen Gadol (High Priest) wore. This garment is made completely of tcheiles. When Korach rebelled, one of the two questions that he asked was whether a four cornered garment made all of tcheiles requires tztzis. In fact, the Minchas Chinuch states that the meil was just such a four cornered garment. The Kohen Gadol was not required to put tzitzis on the meil even though it was required under normal circumstances. This is similar to the avnet (belt) which was made of sha'atnez and would be forbidden in any garment other than the uniform of the Kohen. We see from this that the answer is that any other garment made all of tcheiles would require tzitzis.
It is Interesting to consider the reason that things that are forbidden in "normal life" are required as part of the avodah in the Beis Hamikdash.Other examples are the karbanos (sacrifices) which involve a number of forbidden melachos normally forbidden on Shabbos (such as slaughtering animals).