Matthew 5:33

We have here an exposition of the third commandment, which we are the more concerned right to understand, because it is particularly said, that God will not hold him guiltless, however he may hold himself, who breaks this commandment, by taking the name of the Lord in vain. Now as to this command, I. It is agreed on all hands that it forbids perjury, forswearing, and the violation of oaths and vows, Mat 5:33. This was said to them of old time, and is the true intent and meaning of the third commandment. Thou shalt not use, or take up, the name of God (as we do by an oath) in vain, or unto vanity, or a lie. He hath not lift up his soul unto vanity, is expounded in the next words, nor sworn deceitfully, Psa 24:4. Perjury is a sin condemned by the light of nature, as a complication of impiety toward God and injustice toward man, and as rendering a man highly obnoxious to the divine wrath, which was always judged to follow so infallibly upon that sin, that the forms of swearing were commonly turned into execrations or imprecations; as that, God do so to me, and more also; and with us, So help me God; wishing I may never have any help from God, if I swear falsely.

— Matthew Henry's Concise Commentary (public domain)

Again, ye have heard that it hath been said by them of old time, Thou shalt not forswear thyself--These are not the precise words of Exo 20:7; but they express all that it was currently understood to condemn, namely, false swearing (Lev 19:12, &c.). This is plain from what follows. But I say unto you, Swear not at all--That this was meant to condemn swearing of every kind and on every occasion--as the Society of Friends and some other ultra-moralists allege--is not for a moment to be thought. For even Jehovah is said once and again to have sworn by Himself; and our Lord certainly answered upon oath to a question put to Him by the high priest; and the apostle several times, and in the most solemn language, takes God to witness that he spoke and wrote the truth; and it is inconceivable that our Lord should here have quoted the precept about not forswearing ourselves, but performing to the Lord our oaths, only to give a precept of His own directly in the teeth of it. Evidently, it is swearing in common intercourse and on frivolous occasions that is here meant. Frivolous oaths were indeed severely condemned in the teaching of the times. But so narrow was the circle of them that a man might swear, says LIGHTFOOT, a hundred thousand times and yet not be guilty of vain swearing.

— Jamieson-Fausset-Brown Commentary (public domain)