While there is little that is ironic in the strict sense of the literary term about the narrator's role in "The Fall of the House of Usher," there may be one or two instances to consider.
When the narrator surprisingly receives a letter from a former "boon companion" of his school days, he notes that the sender, Roderick Usher, entreats his "only personal friend" to visit him in order to alleviate his suffering from a strange malady. So, the narrator, who is touched by this entreaty, travels to the house of Usher. After he arrives, the narrator makes efforts to alleviate the melancholy of his former classmate but has no effect on Roderick.
For example, the narrator declares that he "was busied in earnest endeavors to alleviate the melancholy" of his friend. He tries to divert Usher through the influence of literature, art, and music with the expectation that these arts will soothe the soul of Usher as they normally do. However, this expectation is not met.
...bitterly did I perceive the futility of all attempts at cheering a mind from which darkness poured forth upon all objects of the moral and physical universe, in one unceasing radiation of gloom.
We could make the case that this is situational irony because actual events are the opposite of our expectations: we would think a close friend providing entertainment in the form of company, music, literature, and art would cheer someone up, while in the story they only result in continuing melancholy.
The narrator's role is primarily ironic in his relation to the psychic tumult observed. That is to say, when the story opens, the narrator speaks of the gloom and oppression the House of Usher brings into his spirit. It is about him, and seems to be concerning, even obsessing him. The same is true when the story ends; this is a very self-centered narrator. He's concerned about the effects on his own mind and spirit—but the truth is, this is another house/family/person going mad. He's so concerned with keeping his own "house" in order that the deterioration of another's is of relatively little importance.