What central idea is presented in both "I felt a Funeral, in my Brain," by Emily Dickinson and "The Tell-Tale Heart" by Edgar Allen Poe?

This is a great pairing. To me, the thing these two texts share is a descent into madness, or the idea that there is an understanding of things that transcends reason. Both texts share the theme of death: Dickinson is writing about a funeral, and Poe is writing about a murder. Both texts use a kind of monotonous rhythm to highlight this movement from reason to madness. Both texts seem to suggest that once you give into this alternate way of thinking, there is no going back.

In the Poe story, the narrator murders the old man because of his "vulture's eye" -- 

His eye was like the eye of a vulture, the eye of one of those terrible birds that watch and wait while an animal dies, and then fall upon the dead body and pull it to pieces to eat it. When the old man looked at me with his vulture eye a cold feeling went up and down my back; even my blood became cold. And so, I finally decided I had to kill the old man and close that eye forever!

In the Dickinson poem, the funeral inspires a similar kind of dread: 

I felt a Funeral, in my Brain,
And Mourners to and fro
Kept treading - treading - till it seemed
That Sense was breaking through -

And when they all were seated,
A Service, like a Drum -
Kept beating - beating - till I thought
My mind was going numb -

Dickinson's "numbness" and Poe's narrator's "cold feeling" at seeing the eye both suggest a kind of falling away from reason or the real world. For Poe, the case would seem to be one of simple insanity, but it is worth noting that for the narrator the "reasoning" behind the killing is self evident; the eye gives him the shivers, "and so" of course the old man must die. In the Dickinson poem, monotony of the funeral, whether it is an actual funeral she is attending, or a kind of "funereal feeling," brings on a similar quality of "numbness" which results in the "Plank in Reason" breaking

And I dropped down, and down -
And hit a World, at every plunge,
And Finished knowing - then -

The "beating - beating" of the drum in the poem, like the beating of the old man's heart in the Poe story, precipitates a similar fall from knowledge in each text. When Dickinson says she "finished knowing," that can be taken to mean that either she is "finished" with reason, or that she has finally understood something. In the Poe story, the beating of the old man's heart causes the narrator to confess. This confession could be understood as either the narrator giving in to a guilty conscience, or as his final capitulation to the irrational desires that drove him to murder in the first place.