The theme of "Sonnet 55" is one shared by several of Shakespeare's sonnets in his "Fair Youth" cycle (Sonnets 1–126). The theme is that, through writing about his beloved, Shakespeare is able to commit his beauty to paper and thus make it immortal. Compare also "Sonnet 18" and "Sonnet 65" for some examples. The "living record of your memory" referred to in the quoted section, then, is the poem itself, which is a record that will live, according to the poet, far longer than its subject.
To the modern reader, then, there are arguably two forms of irony to be found in these lines: one intended by Shakespeare and one perhaps unintended. The first form of irony—according to the definition of irony as "a state of affairs contrary to what one might expect"—is straightforward. While we might expect the "living record," a poem written upon paper, to be easily destroyed by "war's quick fire" and certainly less enduring than the "gilded monuments" mentioned earlier in the poem, the poet states that this is not the case. This poem, he says, is more durable than marble, which is straightforwardly ironic.
The second potential form of irony is dramatic irony; where the full significance of the words is now clear to the reader, as it may not have been to the speaker. Four hundred years after Shakespeare committed these words to paper, we are still reading them; so, in a sense, he is correct in stating that the "living memory" of his beloved will endure far beyond his lifetime. However, although we know that this young man was very beautiful and beloved by the poet, Shakespeare never gave us his name. So, while he has been the subject of centuries of scholarly discussion, and poems describing his beauty have been read at countless weddings and invaded the public consciousness thoroughly, we do not know who he is. The "living memory" is still with us, but we do not know who we are remembering.