I know of two stories - one in Tanach, one in midrash - in which an angel causes a coal to be touched to a prophet's mouth:
In the year of the death of the king Uziyahu, I saw the LORD sitting on a high and exalted throne, and his robe-skirts filled the hall. ... And I said: "Woe is to me! For I am doomed, for I am a man of impure lips, and among a people of impure lips I dwell, for my eyes have seen the King Hashem Tzevaot." And one of the Seraphim flew to me, and in his hand a coal, taken with tongs from off the altar. And he touched my hand, and said: "Behold, this has touched your lips, and your iniquity is gone, and your sin is atoned." And I heard the voice of the LORD saying: "Who should I send, and who will go for us?" And I said "Behold, I am hear, send me." (Yeshayahu 6:1-8)
Pharaoh would kiss and hug [the child Moshe] and [Moshe] would take Pharaoh's crown and put it on his head, just as he would do in the future when he grew up... The Egyptian magicians were sitting there and said: "We are afraid of this [child] who takes your crown and puts it on his head, lest he be the one who will take kingship from you." Some of them said to kill him [by sword], some said to burn him. Yitro was sitting among them and said to them: "Perhaps this youth cannot [yet] think? Test him and bring before him gold and a coal in a bowl. If he extends his hand to the gold, he can think so kill him. If he extends his hand to the coal, he cannot think and he is not deserving of death." Immediately they brought before him. He extended his hand to take the gold. Gabriel came and pushed his hand and he took the coal and put his hand with the coal in his mouth and his tongue was burned. From this he became "heavy of mouth and heavy of tongue". (Shemot Rabbah 1:26)
It seems clear to me that the midrash about Moshe has its source, in large part, in the story of Yeshayahu's first prophecy. How could such a specific story (an angel causes a coal to be touched to a prophet's mouth) in the midrash not be linked to the same story in Tanach?
The continuations of the two prophets' stories contain another parallel. But this parallel is mirror image rather than identical: God asks who will be "sent" to the people and Yeshayahu immediately volunteers, while God tells Moshe to go to the people but Moshe asks that somebody else be "sent". Yeshayahu is extremely eager to go, while Moshe is extremely uneager.
And finally, there is one important detail that is simply different between the stories. In Yeshayahu, the coal takes an impure person and turns him into a pure person, ready to give prophecy to the people. In the midrash, the coal takes a speaking person and turns him into a speech-impaired person.
What are we supposed to make of the close parallel between the stories, which yet has some important differences?
When a person is touched by coals on the mouth, there seem to be two possible - contradictory - results. Scientifically speaking, this should cause a burn which might lead to some kind of speech defect. Yet with Yeshayahu the burn has a nearly opposite effect: it turns him into a prophet. In a given case, which of these results will occur, and why?
I think the best answer to this comes from Moshe's story:
Moshe said to Hashem: "Please O LORD, I am not a man of words, nor was one yesterday nor the day before, nor since You have spoken to Your servant, for I am heavy of mouth and heavy of tongue." Hashem said to him: "Who gives man a mouth, or makes him mute, or deaf, or seeing, or blind? Is it not me, Hashem?" (Shemot 4:10-11)
Thus sometimes a burn leads to disability, but sometimes - whenever God chooses - it actually leads to a greater ability.
When Yeshayahu realized that God had chosen him as a prophet, he eagerly volunteered - despite the mouth burn he had just suffered. Moshe was placed in the same situation - called to be a prophet despite his limitations. It was only left for the midrash to fill in the exact story of how he got those limitations. By making them the same limitations that Yeshayahu had, the midrash highlights the contrast in their responses.