To reduce the debuff timer you attack Illaoi herself, each instance of damage reducing the duration by one second.
Illaoi is a lane bully by design. Her abilities are meant to prevent you from farming. However, Illaoi becomes less useful after the laning phase.
The best way to win games against her is to pressure other areas of the map (roaming, taking Herald/Dragon/Baron, pressuring other towers). Because she depends on having tentacles near her to significantly zone out the enemy, she's not very likely to roam or teleport anywhere. During teamfights if you either dodge her E or let your tank be hit by it, you have a window to jump her.
In my experience playing Illaoi, I almost always "win lane" but the longer the game goes on the less likely I am to win the game. The passive timer (which is how she summons her tentacles near herself) only gets as low as 12 seconds at level 18, and teammates seem to forget this, starting fights before I can stack up some tentacles around where we plan to fight.
When Illaoi uses her R - Leap of Faith, she summons extra tentacles for each enemy hit, including any spirits currently lingering. Avoid clumping together so she can't R-Flash into your team, and if she does land a strong R, just back away. Every enemy she hits per Slam, she heals herself for 5% of her missing health. She might seem like she's an easy target at low HP, but she's actually stronger.
Overall, Illaoi's main weaknesses are CC (prevents her from using her W to command tentacles), minion blocks (prevents her E from landing), Grievous Wounds (lowers the amount of HP she regains for each tentacle slam), and just killing her tentacles.
I'm sure I'll be kicking myself down the line for exposing my strongest champ like this, but no one knows how to beat a champion better than someone who plays them a lot.
Feel free to add me if you have any more questions about Illaoi or any other champ!
[sg-miss-fortune]