Said about one's self, it was used as a statement of remorse and shame; the idea being that you are watched over by your ancestors, and should never do anything to shame them -- having forgotten the face of your father could be said to be short for forgetting that he is ever watching you. Said about someone else, it indicated they should feel such shame.
Robin Furth (who was a research assistant to King, and wrote the Dark Tower Concordance - an excellent work if you are a fan of the Dark Tower, or King in general) said the following in the Concordance:
To a man who belonged to the tet of the gun, losing his life was preferable to losing his honor, since loss of honor reflected not just on himself but on all of his male ancestors. Among gunslingers, the saying “I have forgotten the face of my father” was the ultimate statement of remorse.
and, later
I HAVE FORGOTTEN THE FACE OF MY FATHER: This is a term often used by Roland. It is a phrase of shame. When one has forgotten the face of one’s father, it means that one has behaved dishonorably. Mid-World is patriarchal, a cultural structure that is older than either the aristocracy of gunslingers (the Barons of Mid-World), or the kingship of Arthur Eld. When Eddie and Susannah visit the Cradle of Lud, they see the sculpted visages of “stern men with the harsh faces of executioners who are happy in their work” (III:343). We do not know whether these men were judges, justices, politicians, or legendary forefathers, but their sculpted faces, somehow reminiscent of busts of Roman senators, tell us something about both the pride and the unrelenting harshness of the Great Old Ones. Like the Romans, or our own culture, they were guilty of hubris. I:97
This then correlates with a later entry:
I SWEAR ON THE FACE OF MY FATHER: I swear upon all I hold sacred. VI:188
And, perhaps the most evocative image:
REMEMBER THE FACE OF YOR FATHER. HE WATCHES YOU FROM THE CLEARING: Remain honorable because your ancestors are watching you. W:285