Back when she was new, Volibear used to be really good against her. Given how items have changed that may no longer apply, but back then Volibear with sunfire cape utterly destroys Illaoi in a slugfest through his various passives.
Gnar utterly demolishes Illaoi.
Jax is alright, but the lane pretty much comes down to who wins the earlier trades and zones the other laner.
Irelia is a skill matchup.
Fiora is good against her. Snowball lane for both sides.
Camille is a skill matchup.
Aatrox can do ok, but he takes a lot of practice.
Tryndamere is a skill matchup with a tough early game.
Taric can brute-force her, but he has massive mana trouble early and takes practice.
Urgot is pretty good against her. The matchup is largely influenced by jungler presence, however.
Garen is really good against Illaoi.
Darius is a skill matchup for Illaoi, which should be easier now that he's been buffed
Renekton and Riven both do really well against her provided they're good at using their spells to dodge through her.
Vladimir pretty much can't die.
Teemo can bully her out of lane if played well.
Pantheon is a skill matchup.
Gangplank can do really well against her, but not explicitly win lane unless she messes up.
Graves used to be pretty good against her.
Olaf is alright against her.
Quinn used to be the designated counterpick to illaoi, but after Illaoi's ult was buffed it's more of a skill matchup. Quinn is still a better bully.
Vayne is a skill matchup, but favouring Vayne.
Rumble is supposedly pretty good against Illaoi.
Trundle is pretty good against everything.
Yasuo is like a cross between Tryndamere and Riven, aka really good against Illaoi provided you play well
Heimerdinger is really good against all melee champions.
Draven destroys Illaoi in lane.