One of my favorite fics, Merging by Shaydrall (unfortunately, it hasn't been updated for more than 6 months), has a premise like this. The most popular solution is that the dementor sucks out Voldemort's horcrux as it is closest to the surface Harry's soul. I highly recommend The Merging.

 

Obviously if a dementor kissed Harry they'd form a soul bond and she'd join his harem.

Talk about obvious.

 

He would be without a soul, Horcrux or not, it would keep sucking till nothing is left. I mean why would it just stop at 1 soul? Can any one stop after 1 potato chip?

 

That is a heavy point of contention, but my theory would be that Nagini was a breed of very long-lived magical serpent, much like the Basilisk. I'd say she'd have a life expectancy of about 500 years, without accounting for the lavish magical healing King Voldemort would have been able to provide for her if he won. If things got dire he could always create a Philosopher's Stone — while he was unwilling to rely on it for immortality at first, I could easily see him creating one as a secondary safeguard, once he had won.

 

I like to think that the Dementor that kisses Harry gets a nice double meal, two souls for the price of one if ya get what I mean.

Like why stop when you've eaten the horcrux? Or why even start with the horcrux? There's a nice big happy soul instead of a deranged, 8th of a soul that the horcrux is.

 

Harry would be dead, Horcrux would take over.

 

Why are we even having this discussion?

It has already been answered by the esteemed fanfiction author Perfect Lionheart in his wonderful fic Partly Kissed Hero. Obviously, Harry would absorb all of Voldemort's memories and become an uber wizard who easily outsmarts and lays the smackdown on all his enemies no matter how strong or experienced they (supposedly) are.