A number of shows have had St. Patrick's Day episodes, though they're not quite as common as episodes centered around other holidays. But whatever, I'll make a list of any that I can remember having seen. Oh, and while the holiday may play a particularly prominent role in many of the episodes listed here, sometimes it's only a very minor aspect of the episode's plot.

30 Rock
Season 6: St. Patrick's Day (Amazon; A.V. Club; IMDb; TV.com; Wikipedia)

Bob's Burgers
Season 10: Flat-Top O' the Morning to Ya (Amazon; A.V. Club; Hulu; IMDb; TV Tropes; Wikia)

Cheers
Season 8: Bar Wars III: The Return of Tecumseh (Amazon; IMDb; TV.com; Wikia)
Season 11: Bar Wars VII: The Naked Prey (Amazon; IMDb; TV.com; Wikia)

The Crazy Ones
Season 1: March Madness (Amazon; IMDb; TV.com)

Home Improvement
Season 7: Desperately Seeking Willow (Amazon; IMDb; TV.com; Wikia)

How I Met Your Mother
Season 3: No Tomorrow (Amazon; A.V. Club; IMDb; OHS Wiki; TV.com; TV Tropes; Wikia; Wikipedia)

Jackie Chan Adventures
Season 2: Tough Luck (Amazon; IMDb; TV.com; Wikia)

The Real O'Neals
Season 1: The Real Lent (Amazon; IMDb; OHS Wiki; TV.com; Wikia)

The Simpsons
Season 8: Homer vs. the Eighteenth Amendment (A.V. Club; IMDb; TV.com; TV Tropes; Wikia; Wikipedia)