Of course, I meant match ups. And yeah, dont say dodge OP, surrender. But to outplay a good jax is difficult. His job is the same as yours in later teamfights, but he will get the early lead in farm.
But I will see the other match ups later, than I will vote.

First, why no sheen it's a critical it should be a item in every Irelia build. Moreover, why ghost ? It may have a lower CD however, flash is still more versatile. Lastly, there is no counter to a AP team comp. Therefore, I have to vote no.

I suggest building Trinity Force when the time has come to. It is a great item but it's not worth rushing anymore. You finish it whenever you actually can without loosing presence. Some games don't last enough to rush it and get away with 0 tankyness. I dunno what you mean with sustain. I know sustain under staying in lane, gaining mana or health and trinity just offers a flat mana pool late (phage]].

I don't think warmogs is a item that is rushable on Irelia as well. It gets effective when you get armor, now that everyone rushes botrk, and then it is a great item but atm Randuin's Omen for example offers way more utility and a good amount of life and armor.

Guardian Angel is something I build if I have the chance to make a use of the revive. I wouldn't put it into the core after the nerfs, just like I wouldn't put Teleport here, even though it's good, because it doesn't synergy with her own kit (no real escape, no burst sustain) and you have to have a team playing around. I'll add it when I rework the sheet below. Covers not all I want it to, thanks for the feedback!

Will come indeed, I'm working as fast as I can and add the champions you might face top but: This is no "counter list", this is a list of match ups, things you can do and personal rating. Jax is indeed a problem and maybe even a counter but it's not impossible to outplay him and it would be nonsense to write "dodge op, surrender" if you know what I mean. I haven't lost too many lanes vs. Jax because I didn't die early and he will always push in order to use his R. And Jax cannot trade with the tower and my jungler later.

The main things are good. But with the 'counter' list I disagree.
Jax with 7/10? Dodge too good vs Irelia, you cant trade.
No Renekton in the list? Or will that come later on?

You suggest building Trinity Force last...

Big mistake; Irelia thrives off of all the stuff it grants her is excellent sustain. Plus, Warmog's Armor is just one of those things you don't get rid off.

Also Guardian Angel every day. It's just the perfect last item for a fighter like her.

This is some amazing stuff you got here, well done!