The command to “shake the dust off your feet” appears only four times in the New Testament. In each case the command is spoken by Jesus to His disciples when He sent them out two by two.
Matthew 10:14-15 NKJV
And whoever will not receive you nor hear your words, when you depart from that house or city, shake off the dust from your feet.  Assuredly, I say to you, it will be more tolerable for the land of Sodom and Gomorrah in the day of judgment than for that city!
Luke 9:4-6 NKJV
“Whatever house you enter, stay there, and from there depart.  And whoever will not receive you, when you go out of that city, shake off the very dust from your feet as a testimony against them.”  So they departed and went through the towns, preaching the gospel and healing everywhere.
In Mark 6:11 Jesus says, “And if any place will not welcome you or listen to you, leave that place and shake the dust off your feet as a testimony against them.”
In the Matthew 10 account, Jesus clarifies His meaning: “Truly I tell you, it will be more bearable for Sodom and Gomorrah on the day of judgment than for that town” (v 15).
Shaking the dust off one’s feet conveys the same idea as our modern phrase “I wash my hands of it.” Shaking the dust off the feet is a symbolic indication that one has done all that can be done in a situation and therefore carries no further responsibility for it. In the scriptural examples, Jesus was telling His disciples that they were to preach the gospel to everyone. Where they were received with joy, they should stay and teach. But where their message was rejected, they had no further responsibility. They were free to walk away with a clear conscience, knowing they had done all they could do. Shaking the dust off their feet was, in effect, saying that those who rejected God’s truth would not be allowed to hinder the furtherance of the gospel. Even the dust of those cities that rejected the Lord was an abomination and would not be allowed to cling to the feet of God’s messengers.
Embedded within this symbolic gesture was the implication that God also saw the dust-shaking and would judge people accordingly. There was a spiritual significance to a disciple of Jesus shaking the dust off his feet. It was a statement of finality about people who had been given the truth and who had rejected it. On their first missionary journey, Paul and Barnabas put Jesus’ words into practice. They had been preaching in Pisidian Antioch, but some of the Jewish leaders of that city stirred up persecution against the missionaries and had them expelled from the region. “So they shook the dust off their feet as a warning to them and went to Iconium” (Acts 13:51). Antioch may not have welcomed the gospel as they should have, but that didn’t keep the message from spreading to other areas. Paul and Barnabas had done all they were sent to do, and the responsibility was now on the shoulders of those in Antioch. The apostles had proclaimed truth boldly. Some had accepted it eagerly; some had rejected it with violence. The apostles were not responsible for the Antiochians’ level of acceptance, only for their own obedience to God.