Basically, Irelia is a lane bully, and a good one at that. In order to beat lane bullies in lane, you have to play a lane bully. I've had success with Urgot, Riven, and Renekton, all of which are stronger bullies due to one thing or another.
Renekton can sustain better while having tremendous base damages.
Riven can out damage Irelia's sustain and well as kill her post 6, which is very difficult to do since the blades heal Irelia.
Urgot can out bully due to ranged damage and armor shred making it a nightmare for her to farm and the 15% reduced damage from his passive helps out a lot too.
I imagine Darius could work well, I've never personally tried it though.
Bullying Irelia and zoning her from farm is the best way to deal with her.