Will American Wii Games Play on an Irish Wii?

Yes. The Wii is region-free for games—any region disc boots on any region console.

Details

• Video Mode: Wii auto-detects NTSC (US) vs. PAL (EU) and outputs correctly via component/HDMI.
• Language: Game text/voice stays English (or whatever the disc contains); system menu language is set in Wii Settings.
• Save Data: Region-free—US saves work on EU console.

Caveats

• Online Services: Nintendo Wi-Fi Connection shut down (2014); some games used region-specific matchmaking—now irrelevant.
• WiiWare/VC: Digital titles are region-locked to the Shop Channel region.
• Accessories: Power adapters differ (US 110V vs. EU 230V); use a voltage converter or EU power brick.

Physical game discs: plug & play.