There is a sense in which characters receive what they deserve in this story. Miss Havisham, as befitting somebody who has taken Estella and made, quite intentionally, a heartless woman, whose job it is to ruin men and hurt them, is actually guilty herself of ruining both Estella's life and also the life of Pip, the "victim" she brings in for Estella to learn her arts on. Her suffering through burning is therefore shown to be a result of her crimes. However, it is important to remember that Pip himself is not entirely innocent, and that his "great expectations" have led him to hurt others and to act in a way that is not true to his character, especially towards Joe and Biddy. To this extent, his punishment is only minor compared to Miss Havisham, but it is important to the novel that he is hurt by the experience:
When I got up, on the surgeon's coming to her with other aid, I was astonished to see that both my hands were burnt; for, I had no knowledge of it through the sense of feeling.
Pip must acknowledge his own crimes and sins in this novel, even though arguably he may be viewed as more sinned against than sinning. His own burning points towards the change in his character and his own greater maturity that he has gained through the experience.
Thank you! This helped me very much!
In addition to what has been said in answer #1, Dickens probably decided to have Pip's arms badly burned for two reasons. One was that it makes it obvious that Pip really tried very hard to save Miss Havisham in spite of the fact that he had good reason to dislike her for deceiving him for so many years. Another reason was probably that Dickens wanted to establish that Pip would be unable to defend himself effectively against Orlick when he walked into Orlick's trap. Here is how Pip describes his pain and helplessness:
Faint and sick with the pain of my injured arm, bewildered by the surprise, and yet conscious how easily this threat could be put in execution, I desisted, and tried to ease my arm were it ever so little. But it was bound too tight for that. I felt as if, having been burnt before, it were now being boiled.
Since Orlick obviously has Pip at his mercy and obviously intends to murder him, the tension in Chapter 53 is intensified. The reader cannot imagine how Pip could possibly avoid being killed and thrown into the lime-kiln to have his body completely consumed. Orlick thinks he has his "enemy" completely helpless, which motivates the cruel man to take his own sweet time about finishing the job. During this tense time, Orlick finally solves the mystery of how Pip's sister came to be badly injured and permanently incapacitated.
“I come upon her from behind, as I come upon you to-night. I giv' it her! I left her for dead, and if there had been a lime-kiln as nigh her as there is now nigh you, she shouldn't have come to life again.