Jesus alluded to this prohibition and spoke of it in secular terms as simply a dumb thing to do:
[Mat 9:16 NLT] 16 "Besides, who would patch old clothing with new cloth? For the new patch would shrink and rip away from the old cloth, leaving an even bigger tear than before.
The word translated "new cloth" refers to wool that has not been "carded". The carding process takes the wool from the form that it is in when you simply cut it off the animal. It is completely and utterly useless as a patch on a garment that has been made of carded wool, linen or anything else.
In this image you can see the uncarded wool and how it looks once carded:
Jesus complains that the uncarded wool patch, as it shrinks will pull away from the old material taking threads/wool with it and making the tear worse. The Pulpit Commentary focuses on the moral instruction of mixing two unlike things together:
...The moral meaning of the whole of this injunction is exhibited in the following passages from the New Testament, "Ye cannot drink the cup of the Lord, and the cup of devils" (1 Corinthians 10:21). "Be ye not unequally yoked together with unbelievers: for what fellowship hath righteousness with unrighteousness? and what communion hath light with darkness? and what concord hath Christ with Belial? or what part hath he that believeth with an infidel? and what agreement hath the temple of God with idols?" (2 Corinthians 6:14-16). "He cannot love the Lord Jesus with his heart," says Hooker, "who lendeth one ear to his apostles and another to false teachers, and who can brook to see a mingle-mangle of religion and superstition' ('Serm.' 5:7, quoted by Wordsworth)
Besides the moral image there is a prophetic dimension to this prohibition in that it communicates that since this is the last day of the temple age and the beginning of the age of the messiah one must not attempt to take the new covenant things and try to patch up or unite with the old covenant.
The next verse is similar, seemingly advising his hearers about proper wine bottling:
[Mat 9:17 NLT] 17 "And no one puts new wine into old wineskins. For the old skins would burst from the pressure, spilling the wine and ruining the skins. New wine is stored in new wineskins so that both are preserved."
If you place wine that is still fermenting into a leather bottle/wineskin the growth will exceed capacity, split the wineskin and the wineskin will be ruined and the wine will be spilled to the ground.
But we know that there is more to the story. The new wine perhaps suggest the holy spirit with all its life and activity that cannot be confined in old covenant, temple-centric Judaism.
In Ezekiel 44:17 these principles are applied particularly to the priests. The priests exclusively wore linen - no wool. This imagery seems to suggest that priest must not sweat during the exercise of his office.
Update
In the comments section on a Youtube video I saw this very relevant information:
"...If you are the person doing the laundry in your household, you will readily understand that a garment woven of wool and linen can never be washed. The tags in our clothes usually say “separate” whites from colors, or in this case wool from linen. Why? Because wool is something you have to wash in COLD water, lest is shrinks. And linen should be washed in HOT water to get it clean and white..."
שַׁעַטְנֵז, which is mixing of wool and linen. Some commentators have said that it was a custom of pagan priests to wear both wool and linen. The prohibition against wool by itself, in your first paragraph, is still sort of a prohibition againstשַׁעַטְנֵז, because the priests were already described as wearing linen. – Tim Biegeleisen Sep 27 '18 at 4:15