The Apostle Paul teaches us to “Owe no man any thing” (Rom 13:8a). Does this mean that it is a sin to be in debt? If it is not a sin, then, in view of Deuteronomy 23:19–20 and Jeremiah 15:10, is it a sin for a believer to borrow or to lend with interest?

No, I do not think that the Apostle Paul is teaching us that it is a sin to be in debt although I know that many do hold to this position (e.g., R.J. Rushdoony). In the first place, the context is not so much about loan-debts as it is about the rendering of tribute, custom, fear and honour (Rom 13:7). But assuming that loan-debt is included in this injunction, in the second place, the Greek (Mêdeni mêden opheilete) may be literally rendered, “to no one, nothing, continue to owe” or “Do not keep on owing anyone anything.” Thus, this command, in fact, assumes the legitimacy of debt, and calls rather that we should not delay to clear our debts when we are able to. Matthew Henry’s judicious comment on this verse is worth noting:

Due payment of debts (v. 8): “Owe no man any thing; that is, do not continue in any one’s debt, while you are able to pay it, further than by, at least, the tacit consent of the person to whom you are indebted. Give every one his own. Do not spend that upon yourselves, which you owe to others.”

We may add that if it is a sin to borrow or to be in debt, then it would,—by implication,—be a sin to lend, and yet we are taught to be generous to lend (Deut 15:7–8, Lk 6:35, etc.).

I believe that every Christian should avoid getting into debt as far as possible, especially when the debt is a result of a covetous desire to extend one’s financial estate or to live beyond the means that God has granted by His providence. One who goes into debt out of luxury or greed would be sinning by way of transgressing the Tenth Commandment. But it is not a sin to be in debt (though we are all debtors to Christ because of sin).

What about charging and paying interest? The two passages you cited are:

Thou shalt not lend upon usury to thy brother; usury of money, usury of victuals, usury of any thing that is lent upon usury: Unto a stranger thou mayest lend upon usury; but unto thy brother thou shalt not lend upon usury: that the LORD thy God may bless thee in all that thou settest thine hand to in the land whither thou goest to possess it (Deut 23:19–20).

Woe is me, my mother, that thou hast borne me a man of strife and a man of contention to the whole earth! I have neither lent on usury, nor men have lent to me on usury; yet every one of them doth curse me (Jer 15:10).

We should note, first of all, that Jeremiah is not saying that it is a sin to lend or borrow on usury (or interest). The most we can infer from his words is that strife and contention often result from lending and borrowing on usury, especially when payment is defaulted. But to read here that it is a sin to lend or borrow on usury would be to read into the text.

Now, as to Deuteronomy 23:19–20, it is true that the Jews were forbidden to lend to fellow Jews (see also Exodus 22:25) on usury although they were allowed to lend to strangers on usury. We may infer from here that Christians should not charge fellow believers interest should we lend to them. This prohibition, of course, only makes sense in the case of a personal loan, as it is practically impossible to enforce in the case of a financial institution, such as a bank. The fact that it is legitimate for banks to charge interest or to pay interest is alluded to in our Lord’s parables which involved the use of the exchangers or the banks (Mt 25:27; Lk 19:23). Note that when we save an amount with the bank, we are actually loaning our money to the bank for an interest!

I should add an additional caveat to lending on interest. Although, Deuteronomy 23:19–20 allows believers to charge interest in the case of loans to unbelievers, we must note that it is only in the case where the unbeliever borrows from us for any other reason than extreme poverty. Thus, for example, if an unbeliever borrows from us in order to increase the capital needed for his business venture, then it is perfectly legitimate for us to charge a reasonable interest or to receive a reasonable interest. If however, an unbelieving friend should be in dire needs, and we decide to give a loan to him, then we must not charge any interest despite the allowance in Deuteronomy 23:20. This is because, when the Law was given, the Jews were never allowed to charge interest to the poor, regardless of whether they were Jews or Gentiles. Moses puts it this way: “If thy brother be waxen poor, and fallen in decay with thee; then thou shalt relieve him: yea, though he be a stranger, or a sojourner; that he may live with thee” (Lev 25:35; italic in KJV). Now, some argue that the words in italics in the KJV (which are not in the original Hebrew) should be rendered “as though, he be,” that is to say: “thou shalt relieve him: as though he be a stranger, or a sojourner.” This rendering is legitimate, but it does not at all remove the need to be compassionate to the poor if they were strangers. In fact, elsewhere, such as in Leviticus 19:10, the Scripture instructs us that the poor strangers should also be well taken care of.

I think the principle is that the rich must never take advantage of the poor. Solomon warns: “He that by usury and unjust gain increaseth his substance, he shall gather it for him that will pity the poor” (Prov 28:8). The same principle instructs us that we must never charge excessive interests even though the person may not be borrowing out of poverty. We ought to be merciful, as God has shown mercy and kindness to us.