Flash: Bit of a no-brainer on most champions.

Ghost: I feel that the mobility Ghost provides is much better on Irelia than other summoner
spells due to Irelia's almost immunity to CC other than knock-ups.

Barrier: Good summoner, but Irelia is not squishy and can hold her own.

Clairvoyance: Hell No!!!

Clarity: Good, but their is much better to choose from.

Cleanse: Great, but the amount of Tenacity you can stack on Irelia makes it pointless.

Exhaust: In my opinion, the only other viable summoner to swap with Ghost.

Heal: Not needed, Irelia is not squishy an has good sustain.

Ignite: Good, but this is more for a pure damage champion.

Revive: You're not Zombie Karthus, don't even take. Hell No!!!

Smite: obvious grab if your jungling.

Teleport: Another viable option to swap with Ghost to go help other lanes to counter gank.

Boots of Swiftness: To aid you sticking to your target. if you have Alacrity Upgrade, at full build you be the fastest thing on the field short of a Janna with Mobility Boots, watch and laugh at anything try and chase your, or try and run from you.

Ravenous Hydra: Lane Pushing Power, Life Sustain, Damage.

Zephyr: Attack Speed to go with Hiten Style, Damage, Cooldown Reduction, More Movement Speed,
Tenacity Stack to go with your Passive and the Tenacious Mastery giving you 60% Tenacity when 1 Champion is near by / 75% Tenacity when 2 Champions are nearby / 90% Tenacity when 3 or more Champions are near by. no one is peeling you off the enemy ADC.

Iceborn Gauntlets: Armour, Mana, Cooldown Reduction, Slow/Nuke Proc.

Spirit Visage: Magic Resist, The remaining 20% Cooldown Reduction, Health, and buff your sustain and received healing effects.

Randuin's Omen: Armour, Debuff to AA and ADC champions, and the AoE Slow Nuke.

All of these items provide extremely high mobility, high sustain, high tank, and really strong nuke potential, and with max Cooldown reduction you can spam your skills preventing you from being peeled off the enemy ADC and the Tenacity to ignore CC.