Ziggs doesnt like champs who get up all close and personal in his face. His escape is not that reliable. Try the likes of Kass, Talon and LeBlanc to wreck his face and deny him farm/kill him. Ward his wraiths and take them as often as possible if he leaves lane to heal.
Scarra has been quoted to say "irelia is now underpowered, never picking her again"
Her sustain has been HAMMERED. If you watch matchups today, irelia is losing lane more and more after her recent nerf. She hasn't lost much midgame to late game power though, being able to tank and jump ad carries.
The hit to her early game was brutal though.
Updated answer to new question:
Irelia has been severely weakened now. If you want hard counters, I'd go with warwick, who beats most tops due to his insane sustain. Olaf counters irelia hard as well, since she has no real way to deal with repeated reckless swings. You could also go untraditional and take vlad top.
For a generic champion, you need to quickly determine
a) if you can outdamage her sustain and outdamage her.
b) you will get any significant jungle help
Quite a few champions can snowball over irelia if they manage to get an early kill. Irelia is really weak early levels so you can fight aggressively here. If you can't straight up outdamage her, engage, bait her hiten style and run away. When she loses her true damage, engage her again.
Best Answer
The main reason shen takes so little damage in trades is because his W skill, Feint, is a shield. Essentially this adds to his hp every time it comes off cooldown. In order to help damage him, you can make sure to hit him as soon as his shield disappears and is on cooldown. It's a bit tricky, but manageable. Damage dealt when his shield is on is reduced by the strength of his shield.
Another reason you might be dealing less damage than you expect is because of the damage reduction effect of his E skill, shadow dash. Taunted champions deal 50% reduced damage.
Also, Shen gains very high HP regen after hitting a unit he has marked with Vorpal Blade. This lets him regain health lost to harassment much quicker than most other champions.
The final reason you might not be dealing as much damage as you want to is shen's defensive items. If shen is stacking armor, then he is also going to take less physical damage. The solution to this is building armor penetration, such as Last Whisper or Brutalizer, or simply taking more armor penetration from runes.