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(Someone else asked a question like this and deleted it while I was writing my answer, so I'm reposting the question.)
The tapestries of the Mishkan were made of wool and linen, as were the priestly garments. These are prohibited as shaatnez. Yet, in the case of the hides used for the covering of the Mishkan, we find that non-kosher types are not allowed to be used. Why here is something treif required?
This really is not unexpected that something that is the choicest material and is used in the service of God is not allowed to be used by normal people for everyday use. For example the blood and fats of animals (Vayikra 3:16-17):
וְהִקְטִירָ֥ם הַכֹּהֵ֖ן הַמִּזְבֵּ֑חָה לֶ֤חֶם אִשֶּׁה֙ לְרֵ֣יחַ נִיחֹ֔חַ כָּל־חֵ֖לֶב לַיהוָֽה׃ חֻקַּ֤ת עוֹלָם֙ לְדֹרֹ֣תֵיכֶ֔ם בְּכֹ֖ל מֽוֹשְׁבֹתֵיכֶ֑ם כָּל־חֵ֥לֶב וְכָל־דָּ֖ם לֹ֥א תֹאכֵֽלוּ׃
The priest shall turn these into smoke on the altar as food, an offering by fire, of pleasing odor. All fat is the LORD’s. It is a law for all time throughout the ages, in all your settlements: you must not eat any fat or any blood.
(See Ibn Ezra ibid.)
While the reason for not wearing shaatnez is not revealed, and may not be because of its use in the Mishkan and priestly garments, the exception can be understood in this context. Furthermore, there is an exception to shaatnez even for a layman in the performance of a mitzva (before Rabbinic prohibition), to put Techeiles made of wool on a linen garment (Yevamos 3b-4a):
לא תעשה גרידא מנלן דדחי דכתיב (דברים כב, יא) לא תלבש שעטנז גדילים תעשה לך
with regard to a prohibition for which one is punished by lashes alone, from where do we derive that a positive mitzva overrides it? As it is written: “You shall not wear diverse kinds of wool and linen together. You shall make for yourself twisted fringes on the four corners of your covering with which you cover yourself” (Deuteronomy 22:11–12). These verses teach that despite the prohibition against wearing diverse kinds of wool and linen, it is permitted to prepare ritual fringes of diverse kinds, e.g., sky-blue dyed threads of wool on linen garments. This shows that the positive mitzva of ritual fringes overrides the prohibition of diverse kinds.
