The context of the reference to Danny Boy is that someone is singing while drunk:
"He drinks a Whiskey drink, he drinks a Vodka drink
He drinks a Lager drink, he drinks a Cider drink
He sings the songs that remind him of the good times
He sings the songs that remind him of the best times
(Oh Danny Boy, Danny Boy, Danny Boy)"
Possibly there is no deeper meaning to the choice of song, other than that it's a well-known song which many people might sing while drunk. Later in the song:
"... He sings the songs that remind him of the best times
(Don't cry for me, next door neighbour)"
This is possibly a parody of "Don't cry for me, Argentina", from the musical 'Evita' by Andrew Lloyd Webber and Tim Rice, which is another well-known song which some people might sing while drunk. (They can't use the exact words because they are under copyright.)