I assume you mean the near-end of the story:

A succession of loud and shrill screams, bursting suddenly from the throat of the chained form, seemed to thrust me violently back. For a brief moment I hesitated -- I trembled. Unsheathing my rapier, I began to grope with it about the recess; but the thought of an instant reassured me. I placed my hand upon the solid fabric of the catacombs , and felt satisfied. I reapproached the wall. I replied to the yells of him who clamoured. I reechoed -- I aided -- I surpassed them in volume and in strength. I did this, and the clamourer grew still.

Montressor had been quiet and in a state of denial regarding Fortunato's motives until then, so the screams reveal his realization that he is indeed the victim of revenge, that his death is near.

Poe uses this passage to show how focused and sadistic his speaker is.  Not only does Montressor scream, but the narrator screams even louder.  Not only does Fortunato get physical revenge on his victim, but he gets verbal revenge as well: his words and screams silence his victim.

The passage reflects the gothic mood of the story: death and decay are inevitable; our shrieks from the grave will never be heard.

How does Montresor's reaction to Fortunato's screams affect the climax's mood in "The Cask of Amontillado"?

As Fortunato's intoxication wears off, he begins to realize he is in a terrible situation. He is chained to the damp wall in a recess of Montresor's family catacombs, and Montresor is walling in him with bricks and mortar. At first, Fortunato moans. Then, as his panic increases and the gravity of the situation completely dawns on him, he begins to scream. As Montresor recounts:

A succession of loud and shrill screams, bursting suddenly from the throat of the chained form, seemed to thrust me violently back.

Montresor first responds to the screams by hesitating and trembling. He draws his rapier in fear and thrusts around. Then he returns to the newly built and unfinished wall behind which Fortunato is screaming. Montresor screams in reply, yelling louder and louder until he "wins" and drowns out his rival. As he puts it:

I re-echoed—I aided—I surpassed them in volume and in strength.

When Fortunato is silent, Montresor completes the wall and moves the bones back in front of it.

Montresor's response adds significantly to the macabre and frightening mood as the story reaches its climax. The screams of the desperate, terrified prisoner and the madman mingling together, louder and shriller, and echoing in the dark chamber, is truly a dramatic finale.