What is the biblical basis against polygamy in the New Covenant? Assuming the Old Covenant is obsolete and fulfilled (Heb 7:18-19), what is the biblical argument against allowing the practice of polygamy?
God's Standard Should Be Our Standard
The biblical standard regarding marriage, from Genesis to Revelation is

One groom, one bride, what e'er betide.

Or in slightly less old fashioned verbiage:

One husband, one wife, for life.

God never sanctioned polygyny in the Tanakh, nor did Jesus sanction it in the New Covenant. God's design from the beginning was stated in Genesis 2:24-25 and reiterated by Jesus in Matthew 19:3-12 and Mark 10:2-12. I teased the following statements from Genesis, Matthew, and Mark:


Marriage between one woman and one man was instituted by God.

The man and woman who marry are fully equal in the eyes of God, but they are different in significant ways. There are obvious similarities and differences, but the differences, in particular, complement each other.

Marriage involves two families, but it is the basis of a third family in its own right. In other words, there is a leaving and a cleaving and a conceiving (though not always).

God joins together two people, one man and one woman, and the two--not three or four or more--become one flesh.

Jesus sanctioned divorce when adultery violates the oneness of the marital relationship, but by the same token Jesus did not command divorce when either husband and/or wife commits adultery. In other words, he held out for the possibility of confession, forgiveness, and restoration of the relationship. Serial monogamy is not the biblical ideal, though Jesus obviously sanctioned (but did not command) remarriage after the death of one's spouse.

Jesus' emphasis on Moses in his discussion with the Pharisees who came to Jesus to test him with their question is significant. According to Jesus, &quot;Moses permitted divorce&quot; because of the hardness of the Israelis' hearts, neither because God had made provision for it in his original plan nor because it was legislated in the Mosaic law.



Is There Any Wiggle Room?
No.

The levirate spoken of in Deuteronomy 25:5-10 (levirate comes from the Latin word for brother-in-law) may seem like an exception to the &quot;one husband, one wife, for life&quot; rule, but it is not. As Constable noted in his explanation of this passage in the NET Bible &quot;Notes&quot;:


&quot;The Israelites were to practice levirate marriage only in cases where the brothers had lived together (v. 5) and the remaining brother was not already married. Living together meant living in the same area, not necessarily residing under the same roof.[286] When another kinsman voluntarily assumed the responsibility of the surviving brother, that brother was apparently under no obligation to marry his sister-in-law (cf. Ruth 4). [my emphasis]
&quot;There were several reasons for this provision. These reasons were the importance of descendants in God’s purposes for Israel, the welfare of the widow, and the demonstration of love for one’s brother (cf. Gen. 38).[287]&quot;


Paul's insistence that a church elder or church deacon must be &quot;the husband of but one
wife,&quot; or as some folks interpret it, &quot;a one-woman man,&quot; cannot be an argument from silence that it's OK for men who do not aspire to be leaders in the local church to have multiple wives. Furthermore, the decision of the polygynous man either to keep or separate from a second or third, or more wives, would be up to the husband, his wives, and the leadership of the local church, with each case judged and ruled upon based on its merits on a case by case basis (see 1 Timothy 2:1 ff.).

A Legitimate Inference Based on Every Example of Polygyny in the Bible
I feel confident in presenting this challenge to any reader of my answer to cite a Scripture passage (not the LDS Scripture, however, as I am not a Mormon) or give me an instance in the Bible in which polygyny was not characterized by familial infighting, deceit, jealousy, strife, violence, resentment, long-standing feuds, favoritism, hurt feelings, spiritual backsliding, or any one of a host of other negative aspects and consequences. There is simply not one instance.
What can we legitimately infer from the overwhelming evidence of negative consequences based on the instances of polygyny in the Bible? The only reasonable answer, at least in my opinion, is that God does not approve of it, and to flaunt God's standards regarding marriage is to court disaster. You can count on it!