Lets say you had 3 wizards, let’s call them A, B and C. If A were to turn B & C into Horcruxes, B were to turn A & C into Horcruxes and C were to turn A & B into Horcruxes. Would killing wizard A destroy the Horcruxes B & C had placed on them or would the Horcruxes remain due to a fragment of A’s soul existing in B & C? Hence making all 3 immortal unless they were killed simultaneously.
Once a body dies, the Horcruxes in it die.
Three wizards who make each other into Horcruxes wouldn’t become immortal. Since creating Horcruxes doesn’t make the body indestructible, their bodies would still be able to be killed.
“Well, you split your soul, you see,’ said Slughorn, ‘and hide part of it in an object outside the body. Then, even if one’s body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged. But, of course, existence in such a form …”
- Harry Potter and the Half-Blood Prince, Chapter 23 (Horcruxes)
By killing Wizard A’s body, Wizard A would become bodiless and Wizard B’s and Wizard C’s Horcruxes in that body would be destroyed, since Horcruxes aren’t able to survive the destruction of their container.
“But my point is that whatever happens to your body, your soul will survive, untouched,’ said Hermione. ‘But it’s the other way round with a Horcrux. The fragment of soul inside it depends on its container, its enchanted body, for survival. It can’t exist without it.”
- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)
Then, killing Wizard B would make Wizard B bodiless, destroy one of Wizard A’s Horcruxes and destroy Wizard C’s last Horcrux, making Wizard C entirely mortal again. Killing Wizard C would then, in addition to killing Wizard C permanently, destroy Wizard A’s and Wizard B’s last Horcrux, leaving both Wizard A and Wizard B mortal. Ripping Wizard A and Wizard B from their bodies wouldn’t have been enough to destroy them, since would be able to create new bodies like the Dark Lord did, but all of their Horcruxes would be destroyed since the bodies housing them were destroyed, so they’d be mortal and able to be killed if they created new bodies.