James Joyce, the great Irish writer, wrote of his book Ulysses that in it he wanted to give "a picture of Dublin so complete that if the city suddenly disappeared from the earth it could be reconstructed out of my book." This alone draws many Joyce aficionados to the Irish capital, where they walk the streets every June (and all year, really) to trace the route of the book's hero, Leopold Bloom. Whether or not you've read the book, there could hardly be a more appropriate place than Dublin for a literary pilgrimage of your own.
Ireland's literary contribution is way out of proportion to the nation's small size. Many of the most heralded writers in fiction, drama and poetry come from Ireland—lest you not be familiar, in addition to Joyce there's Oscar Wilde, George Bernard Shaw, Jonathan Swift, Samuel Beckett, William Butler Yeats, Seamus Heaney, John Millington Synge and Brendan Behan, not to mention contemporary writers Brian Friel, Maeve Binchy and Martin McDonagh.
How did this happen? It's hard to know exactly, but some say it's a byproduct of the island's Celtic culture, in which a lyrical mysticism mingles easily with a hearty oral tradition. Combine that with the many hardscrabble travails of Ireland's people (famine, poverty, subjugation), and voila—experiences both joyful and miserable need to be shared, words go on the page, and generations of stories, poems and plays are born. 
For those inclined to see sights associated with Irish writers, the Dublin Writers Museum seems like a natural place to start. The museum isn't huge, but it's well worth a look. You'll find it north of the River Liffey, just five minutes' walk from O'Connell Street.
You can learn about Yeats and many others when you get to the museum, which also has lots of portraits and busts of authors, books galore (including many first editions) and individual displays on many notable Irish writers. You'll also find unique treasures, such as Samuel Beckett's telephone—which might seem inconsequential unless you're a big fan. Temporary exhibits and, naturally, readings and lectures are also part of the scene; if you're a true literary maven, you'll certainly want to participate.
Literature & History
You can also head to Trinity College, where the Book of Kells, one of the city's best-known treasures, is on display. The Book, a highly decorated Latin translation of the gospels, was created around 800 A.D. by Celtic monks. More historical literary artifacts can be found at Marsh's Library, founded in 1701. The collection is devoted to works of the 16th, 17th and 18th centuries. You can take a tour of the old library; notice the small cells in which library users were locked so they wouldn't steal the books. 
A picturesque Georgian building north of the Liffey holds the James Joyce Centre, of interest to James Joyce fans and the simply curious alike. The museum will appeal especially to those who've read Ulysses or A Portrait of the Artist as a Young Man, but as a matter of course contains a lot of information about the city of Dublin. The center's experience comprises an introductory video, maps, pictures, Joyce-related exhibits and a reading room. In the summer, the center conducts guided walking tours, during which fans can indulge their deepest Joycean fantasies.
Once the sightseeing day is over, visitors can head to the Abbey Theatre for a play, just as they have since W.B. Yeats and Lady Gregory founded it in 1904. Now the National Theatre of Ireland, the Abbey has long been dedicated to nurturing and presenting Ireland's important playwriting tradition, and Brian Friel (Dancing at Lughnasa) and his other Irish colleagues continue to present their latest works.
And when travelers are in need of an appropriate nightspot at the end of their literary day in Dublin, there's Temple Bar, where a prominent pub, Oliver St. John Gogarty's, welcomes revelers of all kinds, among them tourists and locals. You might be interested to know that the pub's founder, Gogarty, was a contemporary of Joyce's, and the great author based one of the characters in Ulysses on him ("Buck Mulligan"). If you're really in the literary mindset, you can think about that as you down your third pint of Guinness.