I'll answer this as complete as I can. It's in the format Ability -- Item.
Compoundeyes -- Wide Lens and Zoom Lens (different percentages)
Early Bird -- Chesto Berry (the berry can only be used once per battle)
Filter/Solid Rock -- various berries can reduce damage from specific types
Illuminate -- Honey
Iron Barbs/Rough Skin -- Rocky Helmet
Levitate -- Air Balloon (until the holder is attacked)
Light Metal -- Float Stone
Poison Heal -- Black Sludge (also Leftovers when not poisoned)
Run Away -- Fluffy Tail, Poke Doll, Poke Toy, Smoke Ball
Scrappy -- Ring Target (on the opposing Pokemon)
Shed Skin -- any status-healing berry (only used once)
Stall -- Full Incense and Lagging Tail (Iron Ball doesn't guarantee moving last)
Stench -- King's Rock and Razor Fang
Sturdy -- Focus Sash
Super Luck -- Razor Claw, Scope Lens, Dire Hit, Lucky Punch, Stick, Lansat Berry (in a pinch)
A few items that are similar to abilities:
If Fire moves would be super effective against you, Occa Berry is the same as Heatproof, but only one use.
Flame Orb/Toxic Orb are able to burn/poison the opponent like Poison Point or Flame Body, if Switcheroo is used.
Berries that cure status in battle are similar to abilities like Insomnia and Limber.
Some berries increase stats "in a pinch" similar to Blaze/Overgrow/Swarm/Torrent.
A few items like King's Rock do parts of what Serene Grace does.
I'll also add that there is no equivalent of Heavy Weight - neither Iron Ball or Lagging Tail actually increase a Pokemon's weight in battle.