I've never been to Ireland, but this was the first thing that popped into my head when I heard of "mathematical tourism":

As the story goes (recounted here among other places), on October 16, 1843, the Irish mathematician William Rowan Hamilton was walking along the Royal Canal in Dublin with his wife, when he invented the basic relation defining the quaternions. (He had previously been thinking about ways of extending the complex numbers to higher dimensions.) Supposedly, he was so excited by this that he carved i=j=k=ijk=-1 into nearby Brougham Bridge, which must have been one of the most spectacularly opaque pieces of graffiti in history. Unfortunately, there is no trace of such a carving now, but there is a plaque commemorating Hamilton's idea.

Licence: JP [CC-BY-SA-2.0 (http://creativecommons.org/licenses/by-sa/2.0)], via Wikimedia Commons

According to the article, since 1989 mathematicians from the National University of Ireland, Maynooth have organized a pilgrimage from Dunsink Observatory to the bridge on the anniversary of Hamilton's discovery. So if you're ever in Dublin in October, you assuredly have someplace to go.

(But be sure not to commute there! :))