To take some liberties with Aristotle's definition of tragedy, Miss Havisham is a tragic character. Possessing a certain hubris, and in her anger at being jilted on her wedding day, she vows to take revenge on men through the instrument of Estella. But, in her arrogance, she becomes ignorant of the fact that if she raises Estella to be cruel to people, it follows that Estella will also be cruel to her. And, in her desire to wreak vengeance upon men, Miss Havisham needlessly harms the innocent Pip.
In her discovery of her sin of pride and her realization of what it has wrought, Miss Havisham has committed an act of injustice--"Oh!...What have I done! What have I done!--against Estella and Pip both, and punishment must be exacted. The fire provides Miss Havisham her punishment. And, since fire is both symbolic of purification and of punishment, Miss Havisham's fall into the fire in Great Expectations carries the significance of a penance for her transgressions. After all, as she repeats " Take the pencil and write under my name, 'I forgive her'!" she has certainly begged for forgiveness.
There is nothing to say for sure that Miss Havisham has tried to set herself on fire. Pip looks in at her, she is sitting close to the fire. As he starts to leave, she catches fire, but he does not say that he saw her move in any way that would have caused her to catch fire.
However, the things that she has been saying show that she feels very bad about the way she has lived. She feels especially bad about the way that she has treated Estella. She feels guilty for the fact that she has taken Estella's heart and left ice in its place.
So we cannot really tell whether she has done it on purpose, but I think you can say that Dickens might have meant this to be her penance.
You can say that the fire cleanses her because it burns the wedding dress and then Pip uses the tablecloth from the feast table to smother the flames.