does Illaoi heal from shaco's clone?

According to the wiki, Illaoi will heal off of champion damage with her tentacles.

The Tentacle damages all non-structure enemies it hits along its length, healing Illaoi for 5% missing health if it hits at least one enemy champion.

According to Shaco's page, his ability Hallucination spawns a pet of target type Champion. Therefore if Illaoi lands a tentacle strike on either Shaco, or his clone, she will heal for 5% missing health. Shaco's clone turns into boxes of target type Minion, which can no longer proc the heal.

Her damaging both the Shaco, the Shaco clone, or a pulled soul won't increase her healing. Separate tentacles hitting will increase her healing, and the more targets the higher likelihood of reaching the range of additional tentacles.

Some tips to duel Illaoi as Shaco: The boxes can fear Illaoi, so using ult at the correct time, you can dodge a tentacle strike with the .5 seconds of untargetability gained from ulting. Therefore the extra damage from clone, along with the .5 seconds of being untargetable, could help you in a duel vs Illaoi. Also, another tip to fight Illaoi, if she ults, back away until the the reduced cool down on her w is expired, or ult to dodge her ult in order to prevent more tentacles from spawning.