Illusions can't use any active ability. Passive abilities that illusions can use:
Evasion
Critical Strike (it should be noted that the red number is before the illusion reduction)
Feedback (only works on melee illusions)
Movement speed (flat and percentage)
Attribute bonus (Strength, agility and intelligence)
Flat HP or mana
True Sight
Radiance burn aura (the aura still will not affect units already affected by a different Radiance aura)
Endurance Aura
Spell resistance from abilities, and base spell resistance
Quelling Blade's Demolish
Passive abilities that illusions can't use (even if the animation might show something different):
Raw damage or armor
HP or mana regeneration
Orb effects (except Feedback on melee illusions)
Cleave
Bash (it works in some rare scenarios, read more about it below), this includes the interrupt on Monkey King Bar
Damage block
Spell resistance from Planeswalker's Cloak, Hood of Defiance or Khadgar's Pipe of Insight
Backstab