I would say the answer hinges on Slughorn's explanation:
'Then, even if one's body is attacked or destroyed, one cannot die, for part of the soul remains earthbound and undamaged.
Harry Potter and the Half-Blood Prince - pp.464-5 - Bloomsbury - Chapter 23, Horcruxes
So you cannot die if you have this undamaged, earthbound anchor. But if that anchor's gone then there's nothing tying you to life anymore, if you don't have a body.
Obviously if you have a regenerated body, that keeps you, hence the regenerated Voldemort doesn't die when all of his Horcruxes are gone, he only dies when his body is destroyed in the final battle with Harry.
However, if he had not regenerated a body he would have had nothing tying him to existence. Part of the soul would not have remained earthbound and undamaged, and he would not have had a body, bye bye Voldemort. Bad time to be you.
I'll go a little further, and consider Dumbledore's views on the subject.
'The seventh part of his soul, however maimed, resides inside his regenerated body. That was the part of him that lived a spectral existence for so many years during his exile; without that, he has no self at all. That seventh piece of soul will be the last that anybody wishing to kill Voldemort must attack - the piece that lives in his body.
Harry Potter and the Half-Blood Prince - p.470 - Bloomsbury - Chapter 23, Horcruxes
'So if all his Horcruxes are destroyed, Voldemort could be killed?'
'Yes, I think so,' said Dumbledore. 'Without his Horcruxes, Voldemort will be a mortal man with a maimed and diminished soul.'
Harry Potter and the Half-Blood Prince - p.475 - Bloomsbury - Chapter 23, Horcruxes
So from this I understand that, as I said above, the body will keep you once all the other Horcruxes are destroyed.
But also, I understand that killing a person with Horcruxes will destroy their body and they'll live a spectral existence because of their Horcruxes. Removing the Horcruxes will leave them with nothing.
If they regenerate a body before their Horcruxes are destroyed, you must remove the body and then that's it. The end. But if they don't have a body to remove when their Horcruxes are destroyed, then I don't see anything keeping them earthbound.
In other words, if they haven't got a body, they are surely in the same position Voldemort is in after his body is destroyed in the final battle with Harry. No body, no Horcruxes, no Voldemort.
Consider: "Without his Horcruxes, Voldemort will be a mortal man" and mortal men need bodies.
My final word, will be JKR's:
Dumbledore had passed the job of destroying them to him, and obediently he had continued to chip away at the bonds tying not only Voldemort, but himself, to life!
One Horcrux remained to bind Voldemort to the earth, even after Harry had been killed.
Harry Potter and the Deathly Hallows - p.555 - Bloomsbury - Chapter 34, The Forest Again