Why does the priest "wash" his hands with the incense three times then takes the smoke and touches the paten and the chalice three times as well?
During the procession of the incense, the people confess their sins before the Lord. That is why when the priest returns to the altar, he prays asking the Lord to accept the confessions of the people as He accepted the confession of the thief on the cross. Therefore, we can say that the priest collects the sins of the people in the censor, as it was performed in the Old Testament when they used to confess their sins while laying their hands on the sacrifice. So here, the people confess their sins on the sacrifice of incense, which symbolizes the Lord. At the time of the Institution Narrative, the priest takes the incense (that carried the sins of the people) and places it on the bread and wine, as the Lord carried our sins in His body—as the Lamb of God who carries the sins of the whole world.
The Consecration Prayer is also known as the Institution Narrative, and is considered the most pivotal point of the Divine Liturgy. The priest censes his hands three times and carries the incense in his hands to the holy bread three times, while saying, "He instituted for us the great mystery of godliness." On the third round of censing, he also carries the incense in his hands above the chalice. After censing the chalice on the third round, the priest also says, "For He was determined to give Himself up for the life of the world." Thus, the Institution Narrative prepares the priest to accept and touch the holy body and blood of our Lord Jesus Christ that will mystically occur through the Consecration Prayer and no longer be bread and wine.