Isaiah’s statement that he is a man of unclean lips corresponds to Peter’s statement that he is a sinful man in Luke 5:8. The purification of Isaiah’s lips correspond to the washing of Peter’'s feet at the last supper in John 13.
Isaiah’s mission was to preach with his mouth and God purifies his mouth with a burning coal. Peter’s mission is be the Vicar of Christ on earth, to preach the Good news to all of creation and so God (in the person of Jesus) purifies Peter’s feet.
And the foundations of the thresholds shook at the voice of him who called, and the house was filled with smoke. And I said: "Woe is me! For I am lost; for I am a man of unclean lips, and I dwell in the midst of a people of unclean lips; for my eyes have seen the King, the LORD of hosts!" Then flew one of the seraphim to me, having in his hand a burning coal which he had taken with tongs from the altar. And he touched my mouth, and said: “Behold, this has touched your lips; your guilt is taken away, and your sin forgiven.” [8] And I heard the voice of the Lord saying, “Whom shall I send, and who will go for us?” Then I said, “Here am I! Send me.” (Isaiah 6:4-8)
Luke puts word’s similar to Isaiah’s “Woe!” statement into Peter’s mouth…
And when they had done this, they enclosed a great shoal of fish; and as their nets were breaking, they beckoned to their partners in the other boat to come and help them. And they came and filled both the boats, so that they began to sink. But when Simon Peter saw it, he fell down at Jesus’ knees, saying, "Depart from me, for I am a sinful man, O Lord." (Luke 5:6-8)
St. John takes up the theme in the fourth Gospel written many years later. In John’s Gospel God doesn’t cleanse Peter’s lips but his feet. As a fisher of men, Peter’s feet come in contact with the earth, symbolic of Peter’s ministry to all the world, so instead of cleansing his lips, John has Jesus cleansing his feet.
He came to Simon Peter; and Peter said to him, “Lord, do you wash my feet?” Jesus answered him, “What I am doing you do not know now, but afterward you will nderstand.” Peter said to him, “You shall never wash my feet.” Jesus answered him, “If I do not wash you, you have no part in me.” Simon Peter said to him, “Lord, not my feet only but also my hands and my head!” Jesus said to him, "He who has bathed does not need to wash, except for his feet, but he is clean all over; and you are clean, but not every one of you." (John 13:6-10)
Both Isaiah and Peter start in the presence of God; Isiaih in a vision and Peter in the boat with Jesus. God asks Isiaih “Who shall I send?” and Jesus asks Peter to “Let down the net”. Both are sent. Both are cleansed, purified and consecrated for their mission. Both recognize that they are sinful and unclean, incapable of carrying out the mission on their own. Both are cleansed; the touching of Isaiah’s lips and the washing of the Apostle’s feet are what consecrates each to their respective vocation.
Incidentally, the fact that God purifies Isiaih and that Jesus purifies Peter is another indication that Jesus is God.
-Tim-