Yes, you need a valid passport to fly from London to Ireland (Dublin, Cork, Shannon, etc.). Ireland is a sovereign state—not part of the UK since 1922 (Irish Free State) / 1949 (Republic). The Common Travel Area (CTA) allows passport-free land/sea movement for British/Irish citizens, but airlines require photo ID for all flights, and non-CTA nationals need passports.
UK citizens lost EU freedom of movement. CTA preserves rights to live/work/access healthcare/vote in local elections in each other's countries—but air travel documentation follows international aviation rules, not CTA informality.
Bottom line: Bring your passport. It's the only document guaranteed to work.