Fateful pain
The pain of this poem is the pain that only fate could bestow. All his life, this man longed to meet his soulmate, and then, once together, they are torn apart forever. Why would fate do this to them? What kind of insight could possibly warrant such torture? This pain is so agonizing that when the narrator meets the lover, the lover is completely perplexed by it, and struggling more and more each day to carry on. The pain of his fate is so heavy to his soul that he questions whether life is even worth it.
The irony of consciousness
Then, one day, he falls asleep next to a magic fountain, and who should be there to meet him in his unconscious but a Greek god—Morpheus, the deity of dreams. Morpheus grants him a vision by connecting his dream to the dream of his lost lover; they get to share a few tender moments together, not in waking consciousness, but in the domain of sleep and mystery. This irony is dramatic in nature, because the lover had never thought to reach out to her in sleep.
The internal, external irony
Where is the external person in the dream? She is in his soul. Where is the Fountain of Love? It's right next to him of course, but in a more serious way, it is inside him. When he discovers the source of love, it is within his own self. When he realizes the sacred mystery of his pain, he discovers that his soul and his beloved's soul are already intertwined forever, and although his daily life might portray them as separate and external, they are intimate and unified in spirit.
The sacredness irony
A person could easily misunderstand this poem by saying, "It's a nice love story." It isn't about two specific lovers, though. This is a sacred myth from the depth of the poet, designed as a response to human suffering and fate. In some senses, the poetry is a thin veil over an existential argument that says that, perhaps the mystery of reality is that we are unified in soul, and separated in our waking realities. The poem is ironically an attempt at "Theodicy," attempting to reconcile the suffering of fate with the love of God (in this case symbolized by Morpheus).
Reality as the myth
When the lover comes back from his mystical encounter with God, symbolized by the union of the archetypes (he unites with his sacred opposite through the arbitration of a Greek god-wizard-person), he is faced with this ultimate irony. Why does he suspect that reality is real, but sleep is only myth? Couldn't it be that the dreaming and sleep are more real, and that waking reality is the myth? This is the irony invoked in the film The Matrix which takes Morpheus's name. It comes all the way from this poem, and before that, from ancient Greek legend.