Like the third beatitude, the “earth” (γῆ) in Matthew 5:13 refers to the land of Israel. If I am right about salt as a preserving agent, then Jesus is telling his circle of disciples they are the ones who will preserve Israel (and not the Pharisees).
The worthless salt is “thrown out and trampled (καταπατέω) under people’s feet.” Although this is the type of thing one might do with worthless salt, there may be a hint of coming judgment on people who do not hear Jesus’s message. Later in the Sermon on the Mount, Jesus warns against giving sacred things to dogs and casting “pearls before swine” (Matt 7:6). This is a troubling verse for many reasons, but on the surface it appears to warn disciples they will be attacked (trampled and torn to pieces) when they preach the Gospel to some types of people. In Luke’s version of the parable of the sower, some seed falls on the path and is trampled (Luke 8:5) In Matthew 13:4 the seed on the path is eaten by birds and Jesus interprets this as the “evil one” snatching away the word of God.
Although the phrase does not appear in Matthew, in Luke 21:24 Jesus says “Jerusalem will be trampled underfoot by the Gentiles” (using the cognate verb πατέω). Revelation 11:2 is a possible allusion to this verse, the Gentiles will trample (πατέω) Jerusalem for 42 months. But Gentiles trampling Jerusalem appears in clear eschatological texts the Hebrew Bible as well. For example, in Daniel 8:13 Daniel asks how long the sanctuary and host will be “trampled underfoot?” Although it is not the same word as Matthew 5:13, the LXX uses a related verb, συμπατέω. Daniel 8 refers to the desecration of the Temple prior to the Maccabean revolt, but Jesus uses the language of Daniel 9:27 to predict the coming fall of Jerusalem (Matt 24:15).
Perhaps Jesus implies a contrast between his (true) disciples and those who are not his disciples. The “not the salt of the earth people” are the Pharisees and other leaders in Jerusalem who are not hearing Jesus nor accepting him as the messiah. After the Sermon on the Mount Jesus demonstrates his authority through a series of miracles (Matthew 8-9), but there are also a series of stories describing resistance to Jesus. The “teachers of the Law” think Jesus is blaspheming when he forgives sin (9:3), the Pharisees complain Jesus is eating with sinners (9:11), the disciples of John the Baptist question Jesus on fasting (9:14) and even John himself wonders if Jesus is really the messiah (11:1-19). Whole villages reject Jesus (11:20-24), the Pharisees condemn Jesus for breaking the Sabbath (12:1-14) and eventually declare his power of demons proves he is an agent of Beelzebul (12:22-37). After refusing to give the Pharisees a sign (Matt 12:38-45), even Jesus rejects his own family in favor of his true followers (12:46-50). By the end of Matthew it is the Pharisees who are judged as blind guides, those who cannot preserve Israel any longer and are in danger of being cast out (Matt 23).
With this overview of Matthew in mind, the saying in Matthew 5:13 may be an encouragement to the disciples to be the preserving agent within Second Temple Judaism and a veiled threat to those who reject Jesus as messiah. That the Pharisees are the ones to be tossed out and trampled is a typical ironic reversal of expectations: those who think they will enter the Kingdom of Heaven will remain outside while others enter the Kingdom before them.
I find this a remarkable warning to contemporary Christianity. There are far too many people who claim to be following Jesus but they are more like the Pharisees. It is very easy for a church or a Christian to become so wrapped up in what people think counts toward religion and piety and completely miss the whole point of following Jesus. This might take the form of religious practices which lose their meaning, or the kind of political activism which mixes a poor understating of the Bible with a radical Americanism. To what extent is Jesus’s warning to those about to be cast out and trampled underfoot a call to the modern Christian church?