Jerome

And taking the body, Joseph wrapped it in a clean linen cloth. Through the simple burial of the Lord, the ambition of the rich is condemned, who cannot even be without riches in their tombs. However, we can understand spiritually and also feel that the body of the Lord should be wrapped not in gold, gems, and silk, but in pure linen cloth; although this also signifies that he wrapped Jesus in a clean linen cloth, who received him with a pure mind.

Commentary on Matthew ↗

Hilary of Poitiers

It was he who wrapped the Lord's body in a clean linen shroud; in this same linen we find all kinds of animals descending to Peter from heaven. It is perhaps not too extravagant to understand from this parallel that the church is buried with Christ under the name of the linen shroud. Just as in the linen, so also in the confession of the church are gathered the full diversity of living beings, both pure and impure.

Commentary on Matthew 33.8 ↗

Thomas Aquinas

There follows the account of the burial: "and having taken the body, Joseph wrapped it in a clean linen cloth." And the reverence and the burial are treated. Concerning the reverence: it was a simple reverence, because it was a simple linen cloth. And therefore, according to Jerome, excessive care in burial is censured. By this linen cloth, according to the mystery, three things are signified. …

Commentary on Matthew ↗