What appears ironic about the room with the woman, child, and fire place compare with Scrooge's accounting and his clerk?
It is hard to answer this question without a more specific description of which room with a woman, child and fireplace you are talking about.
I am assuming that you are talking about the part in Stave II where Scrooge is seeing Belle sitting with her daughter by the fire.
I do not think there is really anything ironic about the comparison. The two scenes are very different, but they are supposed to be different.
In Stave I, Dickens shows Scrooge only giving his clerk a tiny fire. This is supposed to show how miserly and mean he is. The scene in Stave II is supposed to help make Scrooge frealize how badly he's treating people and how much different his life would be if he were nicer. So I don't really think there's any irony there.
Please let me know if you were thinking of some other part where a woman, child, and fire appear.
Irony means the opposite of what we expect to happen in reality; therefore, there is nothing ironic about the room with the fireplace and the woman and her child. It is what you expect from a family, filial love, and the hearth was the center of the family, similar to perhaps the television today. So you see there is no irony in this image.
And you cannot really compare Scrooge's room and the room with the woman and the child, for they are very different.
Irony, I just do not see it in this room that is filled with what it should have, a family that is warmed by its love and the fireplace.
Scrooge's room, however, is a totally different matter. There is much irony in his room, but that is a totally different question.
Ebenezer Scrooge is a miser. His office reflects his lack of luxury and comfort. He is the richest man but he lives poorest and too frugal. He keeps the door of his office open so he can spy on his clerk. He keeps a small fire but makes the clerk keep an even smaller one. He guards the coal box by keeping it by him. The clerk is so cold that he has to wear a cover and even tries to warm his hands by candle.
In Stave 2 the spirit takes Scrooge to a room that is warmed by a fire. It is a small room but it decorated. A child is sitting by a woman and other children are running around the room playing loudly. The father comes in with his arms full of presents. The mother is happy. Everyone is happy. The house is that of his former girlfriend, Belle, whom he had once hoped to marry.
The irony is found in the contrast of the two scenes. Scrooge has enough money to live in beauty with warmth and light. He could well afford a fine office with a big fire to keep both he and his clerk warm, and yet, he lives more poorly than the poor. He lives in self-created misery.
Belle's life has been the opposite. They are not rich but comfortable. The couple’s lives are filled with joy and laughter and children. The home is a warm and comfortable. Belle is happy. The irony here is that Scrooge could have had that life had he made different choices. Money did not bring hi anything more than the dingy office and a fear of spending it.