A typical response to the Irish greeting, "Top of the morning to you," would be "And the rest of the day to you."
According to the book English As We Speak It In Ireland (1910) by P. W. Joyce, the expression was used throughout the country. It may once have been a common salutation used at either end of some small talk, but P. W. Joyce only heard it used ironically or jocularly by Irish people.
"Top of the morning to you" would, like begorra(h) (a minced form of by God), be considered an Oirishism or a Paddyism, something popularly associated with stereotypes of Irishness but which is seldom or never used by Irish people themselves. Learn more here.
Of interest, the thousands of Irish speakers who marched in Dublin recently for their rights weren’t looking for any special treatment. Read more at The Irish Times.