Voldemort couldn't die as long as his horcruxes were intact. What would happen if someone pushed him through the Veil in the Department of Mysteries? Would he die? Or could he move back, freely violating laws of nature? Or would he try to return using his horcruxes despite his body being gone?
Hard to tell.
But I think that it would happen exactly what happened when he was hit by Avada Kedavra: his soul would be kicked out of his body, and would not go through the Veil.
Let's not forget that this room is called the DEATH CHAMBER. I know that this is just a matter of opinion but I think (using details from the text) that what is behind that veil is death itself. This is why Luna and Harry can hear the voices of the dead behind it, they have watched death happen. If Voldemort where to be pushed through the veil, he would litterally be entering death. Having said that, I think that the same thing would occur as when the Killing curse rebounded onto him when he tried to kill Harry: he would still be alive thanks to the horcruxes but would be weak and bodiless. The veil is just another quick way to death, just another Avada Kedavra if you will.
The veil is in the department of mysteries because they do not understand how it works and were researching it. What happens when someone goes through is really unknown. We know they are gone and presumably can not come back but it could be that they can come back and for some reason no one has. But that is the point of the veil. No one knows what happens when you go beyond it.