How are the deaths of Helen Burns and Mrs. Reed similar?

What a great comparison! A knee jerk reaction would be to say, "Not very much!" Helen and Mrs. Reed are very different sorts of people, after all. Helen, one is tempted to say, is a kind of emblem of Christian patience and humility, while Mrs. Reed is vindictive, cowardly, and deceptive. Helen dies in Jane's arms; Mrs. Reed, on the other hand, dies alone in her bed.

I think what is similar is that both women are facing death and trying to come to terms with the meaning of their lives. For Helen, faith is what allows her to face death with a clear conscience; her words to Jane are perhaps the only sincere expression of religious feeling in the book:

I am sure there is a future state; I believe God is good; I can resign my immortal part to Him without any misgiving. God is my father; God is my friend: I love Him; I believe He loves me.”

It is telling that Jane, when she goes to visit Aunt Reed the last time, is thinking of Helen:

"One lies there,” I thought, “who will soon be beyond the war of earthly elements. Whither will that spirit— now struggling to quit its material tenement— flit when at length released?”

In pondering the great mystery, I thought of Helen Burns, recalled her dying words— her faith— her doctrine of the equality of disembodied souls.

Mrs. Reed, however, is not troubled by "equality" of any sort. Her rationale for summoning Jane is to unburden her conscience -- to tell Jane that her uncle had inquired about her, and that she had written to say that Jane was dead. She is not sorry, however. In fact, she blames Jane for her troubled conscience:

Now act as you please: write and contradict my assertion— expose my falsehood as soon as you like. You were born, I think, to be my torment: my last hour is racked by the recollection of a deed which, but for you, I should never have been tempted to commit.

Helen is a central character because she represents "right" spiritual feeling, and through her example Jane is able to make peace with her own inner demons. The "spirit of Helen," as it were, comes to Jane; her recollection of Helen's end gives her the strength to not become vindictive towards her dying aunt. Jane's final words to her Aunt express a hard-earned resignation:

"Love me, then, or hate me, as you will,” I said at last, “you have my full and free forgiveness: ask now for God’s, and be at peace."