In Act 2, Scene 2, Jamie insults Tyrone's "Irish bog-trotter idea that consumption is fatal." In other words, Jamie is accusing Tyrone of the old-fashioned Irish belief that tuberculosis is incurable (and that it is not worth it, then, to seek care for Edmund). Tyrone's response is immediate and angry:
Tyrone:
Keep your dirty tongue off Ireland! You're a fine one to sneer, with the map of it on your face!
Jamie:
Not after I wash my face.
Then before his father can react to this insult to the Old Sod, he adds dryly, shrugging his shoulders.
Well, I've said all I have to say. It's up to you.
Tyrone's response includes an extended metaphor in which O'Neill uses "dirt" to describe Ireland in multiple ways. First, Tyrone accuses Jamie of having "the map of it" on his face. This is a common Irish idiom to describe people who look especially Irish, often referring to freckles that are so dense that they could draw the map of Ireland. Tyrone also accuses Jamie of having a "dirty tongue."
But a "map" is a way of describing the landmass of Ireland; a map of Ireland is a recreation of the "dirt" that makes up the land of the country. Jamie takes this interpretation and turns it into a jab at his father: he won't have the map of Ireland on his face once he's "washed it off." He sees the map of Ireland on his face as dirt, something to be removed. Jamie expresses his hatred for his own Irishness, which he knows will incense his "bog-trotter" father.
But this metaphor is made more complicated by the fact that Ireland has a strong connection to dirt in its national identity. The Irish, in their literature and culture, revere the dirt that makes up the country, describing how it holds history and knowledge from prior generations. O'Neill calls Ireland "the Old Sod" in the stage direction. Ireland is, perhaps more than any other nation in Europe, identified with its physical land—its Old Sod, its Bogland. O'Neill thus intentionally complicates the metaphor of the map written in dirt on Jamie's face by adding a reverent, seemingly affectionate reference to Ireland as "Old Sod."