What does fire symbolize in Great Expectations?

The fire that ignites Miss Havisham's decaying wedding dress and burns Pip's arms and hands symbolizes spiritual enlightenment.

In Chapter XLIX Pip responds to a note from Miss Havisham and travels by coach to Satis House where he finds her sitting in a larger room than her own. She rests in a "ragged chair" before the hearth, close to an ashy fire. After Pip enters, Miss Havisham informs him that she wants to fulfill an earlier request that Pip has made. Then, she inquires if she cannot serve Pip in another way, also. To her question Pip responds,

"Nothing. I thank you even more for the tone of the question. But, there is nothing."

After writing a note on Pip's behalf to Mr. Jaggers, Miss Havisham hands Pip the notepad on which she has just written and asks him to look at the "first leaf [page]" where her name is set down. 

"If you can ever write under my name, 'I forgive her,' though ever so long after my broken heart is dust--pray do it!"

Pip tells her he can sign his name immediately since he, too, needs forgiveness and direction. Then, he asks Miss Havisham about how she came to have Estella as her child; Miss Havisham tells him all she knows. Afterwards, Pip departs, but something tells Pip to assure himself that Miss Havisham is all right before he goes through the gate and heads back. So he glances through her window, and just as he turns to go, "a great flaming light spring[s] up" and Miss Havisham is running and screaming "with a whirl of fire blazing all about her." Pip rushes back to the room and throws his greatcoat over her to smother the flames. He pulls from a nearby table the old tablecloth, too, and finally succeeds in stopping the flames. Finally, Pip holds her until assistance is found. 
It is not until the surgeon arrives and tends Miss Havisham that Pip realizes that his own arms and hands have been burned because the burns are so severe that his nerve endings have also been burned.

Later, around midnight, Miss Havisham wanders in her speech, but in many utterances she asks, "'What have I done!' and then, 'When she first came, I meant to save her from misery like mine.'" After saying these words, Miss Havisham reiterates what she has said to Pip, "Take the pencil, and write under my name 'I forgive her'!" She repeats these sentences in the same order many times before she dies.

Just as Miss Havisham has experienced a spiritual enlightenment through the purification of flames, Pip, too, becomes enlightened spiritually from having suffered the burns on his arms and hands. For, he determines even more that he will assist Magwitch in his escape from England where he is under the penalty of death for having returned. Pip has put aside his repulsion for the old convict, and now appreciates what he has tried to do for him.