Mordekaiser received a rework since the previous answer was posted, which completely changed the way his ultimate works. Previously, it applied a damage over time effect that would create a clone pet of the target for Mordekaiser to command if the target died under its effects, and this could target and affect Shaco's clone.
The ultimate's description now reads:
R - Realm of Death - Cast time: 0.75sec
Mordekaiser slows the target enemy champion by 75% during the cast time, after which they're banished together with Mordekaiser to the Death Realm for 7 seconds.
Units between realms see each other as spirits, considering each other dead and negating any interactions between the two. Only the target and Mordekaiser will enter the realm; other champions cannot follow them. Everything that occurs inside the Death Realm is hidden, and all still inside are killed at the end.
For the same duration, Mordekaiser also consumes the target's soul, reducing their current attack damage, ability power, total attack speed, maximum health, armor, magic resist, and size by 10%, in addition to gaining them for himself. If the target dies while inside, Mordekaiser keeps their partial stats until they respawn.
If one of the two affected champions leave the Death Realm, because of having died or having it dispelled, the other will do so as well, but the stats will remain stolen for the full duration.
Shaco's clone, while considered a "champion" for the purposes of most abilities, is not affected by Mordekaiser's ultimate ability now. Mordekaiser will perform the cast animation, but both he, and the clone, will not teleport to the Death Realm.
Here's a gif, taken from this video, highlighting what happens when this interaction occurs:
Not pictured is that this does put Mordekaiser's ultimate on cooldown, so it is not beneficial whatsoever for Mordekaiser to use his ultimate on a Shaco clone.
Short answer: NO
They are only affected by life steal, even then it has to be an aura like Stark's Fervor, and it only heals the turrets, not you.
Best Answer
According to the wiki, Illaoi will heal off of champion damage with her tentacles.
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.