The street was originally known as Lazers Hill (or commonly Leper’s Hill) or Lazarus on account of it commonly been used as an embarkation area for pilgrims to got to Santiago de Compostela in northwestern Spain. It was later renamed after the Lord Lieutenant and General Governor of Ireland, Viscount George Townsend, in the eighteenth century.
When I was young I heard a different story at school. There was a leper hospital on a street named Misery Hill near where Townsend Street is today and the hospital was known locally as as Lazor Hill, Lazy Hill, Lazars Hill or Lepers Hill. The inmates with the the disease were driven along the streets using forty foot poles - this being the origin of the expression "I would not touch him with a forty foot pole".