The rest of the quote you've referenced is very clear on the consequences of the Dementor's Kiss.
"You can exist without your soul, you know, as long as your brain and heart are still working. But you'll have no sense of self any more, no memory, no...anything. There's no chance at all of recovery. You just - exist. As an empty shell. And your soul is gone forever...lost."
It looks pretty clear that a human without a soul will endure in a coma, a form of persistent vegetative state.
As to the state of the soul within the Potterverse, it seems likely that an otherwise undamaged soul rent wholesale from a body would end up in Limbo, at least until the Dementor was otherwise starved or diminished to the point of nothingness, at which point the soul could then (presumably) travel beyond the veil.