The classic work of Dr. David Daube, “The New Testament And Rabbinic Judaism” identified the phrase as a reference to the traditional Jewish midrash that had become more authoritative than the text of Scripture itself. Much like the “Baptist Faith and Message” or the Catechism of Catholicism, it formed a tradition that invariably made “the word of God of no effect.” Jesus had poured forth his criticism upon that tradition in no uncertain terms (e.g., see discourse on ‘Corbin’) and it should not be surprising that in the Sermon on the mount its contrapuntal use to his own “But I say unto you” forms a landmark hermeneutic of the initiation of the Kingdom of God.
Jesus’ clear reference to Scripture are predicated by “it is written” or “the word of God” and similar unambiguous terms making the phrase “Ye have heard that it was said by them of old time” conspicuous by its obtuseness. But this phrase has been used copiously by biblical historical critics to show a putative dichotomy between the “Law” and “Christ” as if Christ were indicating that his teaching superseded the law it self.
Such claims are understandable coming from an ethos that assumed that Scripture, especially the Old Testament, was at core the production of myth and tradition (e.g., German biblical historical criticism) but has been perpetuated in the teaching of a number of self-identified “open theists.”
The chief problem with the “Law” vs “Christ’s teaching” is that the context of the contrapuntal back and forth is prefaced by:
“Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfill. For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled. Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.” (Matt. 5:17-19 KJV)
That is the exegetical key to understanding what follows. That and the solemn warning that “…except your righteousness shall exceed the righteousness of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven. (Matt. 5:20 KJV).
Jesus was excoriating the “theology of Phariseeism” that had taken up the text of Scripture into its superseding midrash. One can easily think of popular “open theist” authors who use the Sermon on the Mount as a pretext for redaction of the Old Testament to conform to a “kinder and gentler” view of God, or the basis for denying the authority of the law in the age of grace.
As a corollary interest many have claimed that Jesus’s “fulfilling the law” was also a statement to the effect that its purpose has been accomplished and its need now removed. But that is based upon the assumption of a definition that has clear alternatives. In light of the Jesus’s disavowal of coming to destroy the law, the case to be made for “to make to stand” or “to establish by scrutiny and examination, by exegesis and by obedience to the sprit as much as the letter.”  is more consistent with his stated purpose.
All to say that the popular assumption that Jesus was directly quoting from Scripture in his “But I say unto you” does not satisfy a consistent contextual exegesis, and the alternative undermines a number of pretexts for denying a high view of Scripture or enables a valid ground to redact the Old Testament portrayals of God using violence to accomplish His purposes in the Old Testament.
1.) See Guillaume, ‘The Midrash in the Gospels’, Expository Times, 37, 1926, 392.