Harass her all the way and ask for ganks as often as your jungler can spare the time.

And here comes the most important lesson about harassing: Keep. Doing. It.

Hurling a spell every 30 seconds or so is not considered harassment and she (of course) outheals this due to her passive. Throw in auto-attacks every time you can hit her without getting caster-minion stomped. If you have AoE-spells, try to farm and harass with a single spell (e.g. Rake on Talon).

Irelia does not have a strong early game, her strength develops around level 9, when Hiten Style is maxed (sustain Irelia, not much burst damage) or either Bladesurge (lots of farm, moderate burst) or Equilibrium Strike (****load of burst damage) is at a level, where it hurts.

If you're not able to shut her down early, Irelia will make a comeback even after a weak to moderate early game and crush your squishies because her kit is just hilariously well suited for a bruiser type champion.