Dumbledore said that Harry must die, and Voldemort himself must do it. If that were to happen, and Harry was to actually permanently die, how would Voldemort be killed?
If Harry stayed dead the Dark Lord could still be killed if his Horcruxes and body were killed.
By this time, all but one of the Dark Lord's Horcruxes, and with Harry's death the piece of his soul inside Harry, were all killed. Nagini was the last Horcrux remaining, and once she was destroyed, the Dark Lord would be mortal. Harry had no part in her destination, that was Neville Longbottom, so Nagini would have still been killed if Harry had stayed dead. While Harry was the one to confront the Dark Lord in the final battle against him, the Dark Lord was mortal by then, and could be killed by anyone. All it would take at that point, when the Dark Lord was mortal, would be for someone to cast a Killing Curse at him.
Harry's fate after he died (if he stayed dead or not) was largely irrelevant.
The important thing was for Harry to die so that the piece of the Dark Lord's soul inside him would die. Dumbledore telling Harry he needed to die was because he would otherwise be trying to defeat the Dark Lord without dying (either temporarily or permanently) and if he succeeded then the Dark Lord would still have an intact soul piece and wouldn't be truly defeated. Killing that soul piece was the entire point of Harry having to die before the Dark Lord could be defeated.
If Harry lived, once the soul piece died, that was fine - Harry himself didn't need to be dead.
The only reason Harry had to die was because it was the only way to kill the piece of the Dark Lord's soul inside him. Harry himself being dead didn't matter.
If Harry stayed dead, then the soul piece would also be dead, so mission accomplished.
The important part was Harry dying so the soul piece would, not Harry coming back to life afterwards. In the bigger picture of orchestrating the Dark Lord's downfall, the most important thing was making sure all of his Horcruxes were destroyed. If Harry stayed dead, the most important part of him dying, the soul piece being killed, would still have succeeded. It worked out so that by the time Harry dies, all the Horcruxes except Nagini were destroyed. Nagini wasn't an object, she was a venomous snake controlled by and loyal to the Dark Lord, so even people who were unaware that she was a Horcrux would be trying to kill her. Neville, the one who killed Nagini, had no idea she was a Horcrux. The last remaining soul piece was in the Dark Lord himself. Both of them were clear and obvious targets to try to kill.
If Harry had died permanently then Voldermort would have become master of the elder wand. This would have made Voldemort pretty much unbeatable so the fact the Horcruxes where destroyed would not really have mattered.
Re reading this part of the books I realise I am wrong here, Voldermort would never have mastered the Elder wand and in dying Harry would have ensured no one else held the elder wand. But the fact is still that even with the Horcruxes destroyed Voldermort was still a far stronger wizard. Remember he only lost the last wizarding war because of Harry, if he hadn't gone to kill Harry he would not have been defeated. Therefore it was only a matter of time before Voldermort could have conquered the wizarding world.
Harry had to die to destroy the Horcrux he had become but he then had to come back to life so he could reclaim the elder wand, remember Voldermort died because the wand betrayed him turning his own killing curse against him. So if Harry had died then I don't think Voldermort would have been killed, remember during the first war Voldermort almost won. Armed with the elder wand he would have created new Horcruxes and having learnt from his previous mistakes probably been unbeatable.