What if, what if, the boy knew about the others? Could he know, had he already acted, had he traced more of them? Was Dumbledore at the root of this? Dumbledore, who had always suspected him; Dumbledore, dead on his orders; Dumbledore, whose wand was his now, yet who reached out from the ignominy of death through the boy, the boy -
Harry Potter and the Deathly Hallows, Chapter 27, The Final Hiding Place
He then goes through each of his Horcruxes in his mind to deduce which ones Dumbledore might have suspected or discovered.
Which should he visit first, which was in most danger? An old unease flickered inside him. Dumbledore had known his middle name...Dumbledore might have made the connection with the Gaunts...Their abandoned home was, perhaps, the least secure of his hiding places, it was there that he would go first...
The lake, surely impossible... though was there a slight possibility that Dumbledore might have known some of his past misdeeds, through the orphanage.
Harry Potter and the Deathly Hallows, Chapter 27, The Final Hiding Place
By this time Voldemort has broken into Dumbledore's tomb and acquired his Elder wand. Surely his immediate reflex would be to perform Priori Incantatem to do a quick check, instead of wasting time assessing the weakest and most vulnerable locations. Also, it would be more reliable information than guesswork. He would have found out instantly that Dumbledore tried to get to the ring and the locket, by penetrating the defenses of the Gaunt shack and the cave, and would have known where to go check first.
Also, even without the Gringotts break-in, he must have definitely performed the spell out of curiosity, given that,
Dumbledore was his greatest enemy and was the leader of the anti-Voldemort movement. So performing Priori Incantatem on his wand would reveal a great many of his secrets to Voldemort.
The Elder wand being the most sought-after and the most powerful wand, and that Voldemort spent almost a year trying to find it, there must be a certain curiosity about the wand's past deeds, even if it weren't Dumbledore's.
Performing Priori Incantatem on a known enemy's wand seems like a natural thing to do, as seen from the following quote:
"But he'll know soon, if he doesn't already, that mine's broken beyond repair," said Harry quietly.
"No!" said Hermione, sounding frightened. "He can't know that, Harry, how could he -?"
"Priori Incantatem," said Harry. "We left your wand and the blackthorn wand at the Malfoys', Hermione. If they examine them properly, make them re-create the spells they've cast lately, they'd see that yours broke mine, they'll see that you tried and failed to mend it, and they'll realize that I've been using the blackthorn one ever since."
Harry Potter and the Deathly Hallows, Chapter 24, The Wandmaker
And we know he didn't because, if he had, he would have found out about the Horcrux hunt.
So, considering all these factors, why didn't Voldemort use Priori Incantatem on the Elder wand?