There are many opportunities for research in medieval and early books in Dublin:
The UCD Irish Franciscan Archives hold Irish and Continental manuscripts from the eleventh century, as well as a valuable collection of early printed books.
Trinity College Dublin Manuscripts and Archives holds a large collection of medieval manuscripts from across the world and Trinity College Library Early Printed Books houses early printed books. See also Beyond the Book of Kells series
With a collection of over 25,000 early books, Marsh’s Library is an eighteenth-century library preserved in its original state.
Edward Worth Library, Dublin is the collection of an eighteenth-century Dublin physician which specialises in medicine and early science, but also holds a fine collection of early modern typography and book bindings.
Royal Irish Academy holds a number of early Irish manuscripts, an Arabic medieval astronomical tract and a French Book of Hours among other holdings.
Located on the Grounds of Dublin Castle, Chester Beatty is a delightful collection in a beautiful space. It houses a diverse collection of manuscripts, early print books,
art works and other artefacts from across the world.
Within commuter distance of Dublin in the university town of Maynooth (easily accessed on the DART) is the Russell Library which houses a collection of 34,000 books including incunabula and a small collection of manuscripts. (For Manuscript collection see the Lucas’s Catalogue of Medieval Manuscripts.) The Maynooth Library also recently acquired the St. Canice Cathedral Collection.
Most of the sites will give you directions, but if you are planning your trip to Ireland this overview map might be of use.