When Isaiah arrives at God's throne, he is commissioned to go speak to the people of Judah and Israel. Isaiah says that he can't because he is "a man of unclean lips" plucked from "a people of unclean lips." An angel picks out a burning coal and presses it to Isaiah's lips. Despite needing an immediate application of Vaseline to his kisser, Isaiah is now able to prophesy. The voice of God can speak through him, bringing his message to the people of Judah.
So, essentially, the hot piece of coal represents pain's ability to purify and cleanse. The Suffering Servant undergoes pain in an even more intense way, which also helps to purge people's sins. In this case, Isaiah's pain doesn't purify anyone else, but it does help him to give out the good (and bad) news.
The poem "Vacillation" by William Butler Yeats uses the coal as a symbol of a higher spiritual reality that burns away our consciousness of the usual, everyday material world. In the poem, Yeats' soul tells him that a human can desire nothing more than to be burnt by that fire, but his heart tells him that he'd rather continue talking about the everyday world, asking "What theme had Homer but original sin?"