The significance of the angel of God touching Isaiah's mouth with burning coal was to

When Isaiah saw the Lord of hosts on his throne, he felt unworthy and recognized his own sinfulness. The angel of God took a burning coal from the altar and touched Isaiah's mouth with it, saying "your guilt is taken away, and your sin atoned for." The significance of this act was to symbolize the cleansing and forgiveness of Isaiah's sins, enabling him to be worthy and ready to accept the commission that God had for him. By purifying his lips, the angel was preparing Isaiah to be the Lord's messenger to speak to the people on God's behalf.