Here's a few tunes (some are reposts). Any additions are welcome.

Danny Boy (F Ballad) - Traditional
Danny Boy (F Swing) - Traditional
How Are Things In Glocca Morra (F) - Burton Lane-Yip Harburg
Irish Washerwoman (C) - Traditional
My Wild Irish Rose - Chauncey Olcott
When Irish Eyes Are Smiling - Ball-Olcott-Graff