What does the speaker in "Ode to a Nightingale" want to forget?

The speaker, listening to the beautiful song of the nightingale, wants to forget all the problems that go with human consciousness. Keats would like to fly away and be amid the beautiful flowers with the nightingale. He wants to forget about such situations as stress, disease, old age and despair. From his perspective, the nightingale does not ever experience anxiety about such human problems as:

The weariness, the fever, and the fret
Here, where men sit and hear each other groan;
Where palsy shakes a few, sad, last gray hairs,
Where youth grows pale, and spectre-thin, and dies

Although the speaker has mixed feelings about death—he says he is "half in love with easeful Death" and feels it would be lovely to die while listening to the intoxicating nightingale's song—he nevertheless contrasts the human consciousness of mortality to the nightingale's oblivion as it lives in the present moment. He calls it an "immortal" bird because it stays the same throughout history. It lives just as it did in Biblical days, singing the same sweet song without forethought about the future.

To become a nightingale represents, to Keats, leaving behind the pain that human consciousness brings.

Why does the speaker in "Ode to a Nightingale" want to "dissolve" and forget?

The poem "Ode to a Nightingale" is about a man lamenting feelings and experiences he wishes to forget using a nightingale's song as an anchor of peace and hopeful respite. The speaker wants to "dissolve" and "quite forget" his own mortality and the existential torment that comes with the thought that living only leads to one thing—death.

In the beginning of the first stanza, the speaker feels terrible, as if he had ingested a deadly poison, but by the end of the stanza, we come to find out that this pain he feels is intensified by the nightingale's song as the bird appears, "too happy in thine happiness," while it "Singest of summer in full-throated ease."

Yet this melody seems to comfort the speaker in a dark way inspiring him to "drink, and leave the world unseen, And with thee fade away into the forest dim." It seems the nightingale's song is paradoxical. It soothes him, giving him the immortality he craves, yet he wants this immortal sound to carry in him into an infinite sleep so that he, too, may become immortal.

In stanza three, we get to the crux of your question, as the lines read:

Fade far away, dissolve, and quite forget

What thou among the leaves hast never known,

The weariness, the fever, and the fret

Here, where men sit and hear each other groan;

Where palsy shakes a few, sad, last gray hairs, Where youth grows pale, and spectre-thin, and dies; Where but to think is to be full of sorrow And leaden-eyed despairs, Where Beauty cannot keep her lustrous eyes, Or new Love pine at them beyond to-morrow.

Here, we find the depth of the speaker's existential woes, for he shares the future that all men face—aging and death. He then speaks of nature as if nature doesn't also die when he says the "leaves hast never known." We see this same sentiment later in the poem when he says the bird "wast not born for death ... No hungry generations tread [it] down."

While the concept of death is, in fact, true for all of nature, the difference rests in the consciousness of man. Man must live life knowing he will surely die which makes the comfort of the nightingale's song all the more enchanting and joyous. It pulls the speaker out of his pain and despair and gives him solace for a short time; the nightingale is never wiser for the knowing.

It seems the bird and the bird's song relieves the speaker of this tormented reality, a reality the speaker contemplates "leaving" in various ways when he says things like, "I have been half in love with easeful Death." Ultimately, the speaker wishes to dissolve the pain he feels and forget the negative experiences and feelings that have lead him to these suicidal ideations through the bird's song which in turn, fuels his existential fire.