Interpretations
The metaphor seems to be teaching against giving what is holy to those who do not appreciate it. Animals such as dogs and pigs cannot appreciate religion, and this verse implies that there is some class of humans who cannot either. What this class is is a difficult question, as one of the dominant ideas in Christian thought is universalism. This verse also seems to contradict what had just gone before. Jesus has just spent five verses attacking the judging and condemning of others, but in this verse states that some section of the populace can be judged and found equal to dogs and swine.
Historically the most common view was that this verse refers to the Eucharist. Only those in a pure state can take communion. Modern scholars reject this approach as the ritual of the Eucharist developed long after the gospel was written, and nowhere else does Matthew make reference to it. Some early Christians believed that some parts of Christianity, such as the secrets of the Eucharist, should be kept secret from all but a chosen few. This followed the hierarchical patterns of the various mystery cults that were popular at the time in the Roman Empire.
One modern argument is that dogs and pigs represent Gentiles and heathens, and that this verse is rare relic demonstrating that Jesus' original message was intended only for the Jews. Harrington notes that such warnings are found in rabbinic works of the period. In Jewish literature heathens were often compared to dogs, and the unclean pig was a Jewish symbol for the Roman Empire. In 2 Peter 2:22 dogs and swine quite clearly refers to heretics. According to Schweizer this verse was used by Jewish Christians to attack the Gentile churches, to argue that Gentile Christians would turn on the Jews by rejecting their laws and destroying Israel.
The dominant reading is that the two expressions are both referring to the same thing and the same group of people. To Nolland this verse is not an attack on any particular group, but rather a continuation of the theme of God and Mammon begun at Matthew 6:24 and that verse is an attack on wasteful spending. We should put all of our resources to God, as everything is like dogs and pigs compared to Him. Nolland also proposes that the verse might be to balance the other verses, that non-judgmentalism can only go so far and that there are some who should be excluded.
As Morris points out, this verse can also been read as a reasonable limit on evangelism. If a population or individual is not open to Christianity, leave and find a more receptive audience. As Morris points out Jesus was silent before Herod and Peter abandoned the unsympathetic city of Corinth. Fowler links this to the earlier discussion of judgment. One should not judge severely, but there is a point at which any reasonable person will realize that those they are dealing with are dogs and swine.
The alternate interpretation is that dogs and pigs are not metaphors for some group of people, but for the unholy in general. This verse is not about excluding some group from God's teaching, but rather ensuring that those things that are God's are kept holy. Thus the Temple is kept clean, religious meals treated with respect, and holy days honoured and kept separate from the turbulence and impiety of daily life.
Read more about this topic: Matthew 7:6