To continue from my last blog, I want to argue that Paul seems to assume that the teachings of Jesus pertain to the New Covenant era.
The Law of Christ
In Galatians 6:2, Paul exhorts us to bear one another's burdens thereby fulfilling "the law of Christ". This law of Christ probably is to be identified with Jesus' teaching to love one another (see also 1 Corinthians 9:21, where Paul speaks of himself as "not God's lawless one, but Christ's enlawed one"). Evidence if this comes from Galatians 5:14 which echoes Jesus' teaching in Matt 22:34-40/Mark 12:28-34. The "loving one another" is explicitly cited by Jesus in John 13:34; 15:12, 17 as a command, which is of course related to the idea of a law. As a side note, in John 13:34 this command is called "new" and Raymond Brown has argued that this is related to the New Covenant. Furthermore, the new covenant passage in Jeremiah 33 speaks of a law being written on the heart. This connection would bolster the hypothesis that "the law of Christ" in Paul is definitely a new covenant idea--an idea going back to the teachings of Jesus. It is not an accident that much of Jesus' sermon on the mount has to do with one's heart and not merely outward appearance. This provides a nice segue to my next topic.
The Word of Christ
In Colossians 3:16 Paul exhorts to "let the word of Christ dwell in you richly". It's possible that the "word of Christ" concerns that message about Christ, but a case can be made it refers to the Message of Jesus, that is, the word spoken and taught by Jesus. Earlier in Colossians we read (3:12-14):
As God's chosen ones, holy and beloved, clothe yourselves with compassion, kindness, humility, meekness, and patience. Bear with one another and, if anyone has a complaint against another, forgive each other; just as the Lord had forgiven you, so you must forgive. Above all, clothe yourselves with love, which binds everything together in perfect harmony.David Wenham has pointed out that this passage has similarities with Matthew 5:48/Luke 6:35, 36. This is evidence that the "word of Christ" in 3:16 is related to Jesus' teachings.
Theological Upshot
The gambit to claim that Jesus' teachings are obsolete flies in the face of too much New Testament, that is, post-advent-of-New-Covenant material. I find it interesting that what many find troublesome about Jesus' teachings in the Gospels are repeated by Paul in his letters! This is a strong indication that evangelicals are getting Paul wrong.