John Chrysostom

Here again, he does not mean flesh absolutely, but such sort of flesh, that which was in a whirl and thraldom of passions. Why then, it may be said, does he not say so, nor state any difference? It is to rouse the hearer, and to show that he that liveth aright is not even in the body. …

Homily on Romans 13 ↗

Theophylact of Ohrid

Furthermore, "you are not living according to the flesh," that is, you do not serve the fleshly life, but the spiritual. Why then did he not say: you do not live in sins? So that you would know that Christ not only extinguished the tyranny of sin, but also made the flesh lighter and more spiritual. Just as iron, from prolonged contact with fire, itself becomes fire, so too the flesh of those who have received the Spirit through baptism entirely becomes spiritual. …

Commentary on Romans ↗

Irenaeus

But we do now receive a certain portion of His Spirit, tending towards perfection, and preparing us for incorruption, being little by little accustomed to receive and bear God; which also the apostle terms "an earnest," that is, a part of the honour which has been promised us by God, where he says in the Epistle to the Ephesians, "In which ye also, having heard the word of truth, the Gospel of your salvation, believing in which we have been sealed with the Holy Spirit of promise, which is the earnest of our inheritance." This …

Against Heresies Book V ↗

Gregory the Dialogist

In Holy Writ flesh is named in one way according to nature, and in another way according to sin or corruption. For there is flesh according to nature, as where it is written, "This is now bone of my bones, and flesh of my flesh." And, "The Word was made flesh, and dwelt among us." But there is flesh according to sin, as where it is written, "My Spirit shall not always abide in those men, for that they are flesh." And as the Psalmist saith; "For He remembered that they were but flesh, a wind that passeth away, and cometh not again." …

Morals on the Book of Job, Book 14 ↗

Thomas Aquinas

Then when he says, "but you are not in the flesh," he shows that those to whom he is speaking are immune from the prudence of the flesh. In regard to this he does three things. First, he describes the state of believers, saying, "but you are not in the flesh." This makes it clear that he is not speaking about the nature of the flesh. For the Romans, to whom he was speaking, were mortal men clothed in flesh. Rather, he is taking flesh for the vices of the flesh: "flesh and blood cannot inherit the kingdom of God" (1 Cor 15:50). …

Commentary on Romans ↗