When Dumbledore died, those who knew the secret became Secret-Keepers, including Snape. Why didn't Snape reveal this information to Voldemort and/or his fellow death eaters?
Wouldn't it arouse suspicion if Snape knew the secret, but refused to reveal it?
I think the correct answer is that it is a plot-hole, plain and simple. To answer the defenses:
Moody's Tongue Tying Curse: First of all, Snape needn't have revisited Number 12 in order to reveal its location to Voldemort. Anyway, even if Snape's tongue was tied when he tried to speak of the location, Snape could have just as easily written out the address on paper (just like Dumbledore did for Harry in OotP).
No need to know the location: Of course Voldemort would want to be able to check out the location, especially because of the fact that Death Eaters were waiting outside for the location to be revealed as answered in previous responses. Why would Voldemort (or other Death Eaters for that matter) wait around until they made a mistake? Surely, either Voldemort or the other Death Eaters would have just asked Snape?
The Death Eaters didn't know Snape knew of the location: Voldemort knew that Snape was dealing with the Order, Voldemort tried to get rid of the Order and, of course, get to Harry, and therefore would have let the Death Eaters know the hideout.
Snape just lied to Voldemort: No, I think even though Snape was a very good Occlumens, he is still no match for Voldemort. Really, the only reason Voldemort let him live after he came back was to tell Voldemort the truth (on Dumbledore's orders). If he simply lied to Voldemort, I think this would be a plot-hole in itself.
Voldemort didn't know that Snape would be able to reveal the location: First of all, the Fidelus Charm is very touchy and very hazy with regards to how it works. For instance, one simple related problem includes, why would Bellatrix not be able to know where it is for surely she had been there before the Fidelus Charm was placed? But the flaws in that aside, as the poster replied to a previous response, Voldemort knows how in works.
"Snape just lied to Voldemort: No, I think even though Snape was a very good Occlumens, he is still no match for Voldemort." What makes you say this? Remember, Snape spent years working as a double-agent against Voldemort. If he could conceal this fact from Voldemort, why couldn't he also hide that he knows the location of Grimmauld Place?
We know that Mad-Eye Moody (a wizard of considerable skill) placed a spell on Grimmauld Place meant to prevent Snape from revealing its location:
“But surely Snape will have told the Death Eaters the address by now?” asked Harry.
“Well, Mad-Eye set up a couple of curses against Snape in case he turns up there again. We hope they’ll be strong enough both to keep him out and to bind his tongue if he tries to talk about the place, but we can’t be sure. It would have been insane to keep using the place as headquarters now that its protection has become so shaky.”
—Harry Potter and the Deathly Hallows
Presumably there were spells that did more than literally "bind his tongue," since we know from Order of the Phoenix that the secret can be revealed through writing.
As long as Snape showed up, he should have been both prevented from entering and from speaking its location.
We know he went to Grimmauld Place:
And next, Snape was kneeling in Sirius’s old bedroom. Tears were dripping from the end of his hooked nose as he read the old letter from Lily.
—Harry Potter and the Deathly Hallows
Of course, the spells clearly did not succeed in keeping him out, since he managed to get to Sirius's bedroom. However, there would be no reason for Snape not to deliberately trigger the Tongue-Tying Curse. By doing so, he could ensure that there was no way he could reveal the information to Voldemort, even under torture, the Imperius Curse, Veritaserum, or some other form of compulsion.
Snape was one of the best Occlumens around, capable of lying to Lord Voldemort's face without getting caught.
“But he’s a very good Occlumens, isn’t he, sir?” said Harry, whose voice was shaking with the effort of keeping it steady. “And isn’t Voldemort convinced that Snape’s on his side, even now?
—Harry Potter and the Half-Blood Prince
What would Snape have claimed? Why, something very close to the truth. He would claim that he had been to Grimmauld Place and been the victim of Moody's spell, which Voldemort could possibly confirm. All he would need to conceal would be that he had done this intentionally. If Moody's curse had been insufficient, he could have placed his own, or perhaps even lied to Voldemort about the scope or nature of the curse. Even unto the end, Voldemort trusted Snape (understandable, given what Snape had sacrificed for him). Further, his supreme confidence in his own powers of Legilimency would have lulled him into believing any information gleaned from Snape's mind.
Moody had placed a tongue tying curse on Professor Snape so that whenever he tried to reveal the secret location he would not be able to speak. Also he had set up traps against Professor Snape in case he ever tried to come back to the Order Headquarters. I'm sure he could have easily fought against these spells, but he probably lied to Voldemort and others about not being able to reveal the location (which is not explicitly mentioned, but can be guessed from previous information)
EDIT:
By this point Snape was one of the most trusted members of Voldemort's gang, and also, it should be noted that Voldemort does not share much information among the members of his own group. It's possible others didn't even know about Grimmauld Place, or the fact that it was Order headquarters or that Snape was now one of the secret keepers. We know from Snapes memory in the last book that Snape came looking for things in Grimmauld Place shortly after the end of book six when he found the picture of Harry on his toy broomstick. This search might even have been done on behalf of Voldemort's orders to make a final check on what was left in the Order headquarters, after which Snape would, rightfully, report that there was not much to see in there any more (other members of his gang wouldn't even have to know that he sent Snape on such an errand). Since he trusted Snape, after this Voldemort would not have any reason to think about Grimmauld Place any more.
This is just a speculation of mine and I don't have anything to back that up:
Maybe he did reveal it.
After Dumbledore's death, the Order was temporarily dismantled and moved from 12 Grimmauld Place. Snape could have revealed the OoP headquarters to Voldemort, with him, Snape and some other Death Eaters visiting it. But this reveal could happen after Snape knew that the Order had completely removed anything that could be used against them (maybe through Mundugus, we know that he used the Imperius Curse on him). Voldemort found nothing after their visit and left the place.
As for the taboo jinx: if we take into account that this jinx could not break the Fidelius Charm, the Death Eaters that were gathered outside 12 Grimmauld Place just knew that someone spelled Voldemort's name but they couldn't find the particular place or suspect that was Harry Potter and his company. And remember: the majority of them were Snatchers not Death Eaters; they didn't answer directly to Voldemort, so they couldn't possibly know that around this place were the headquarters of the Order. I can assume that Voldemort wouldn't inform all his followers about the particular location of it, let alone the fact that Snape should be the one to reveal the location to others.
This of course does not explain why Voldemort himself haven't visited again 12 GP to see if Harry Potter was hiding there, especially after his failed attempt to capture him at Bill and Fleur's wedding. But, considering that Snape worked for Harry and the trio had at all times the portrait of Phineas Nigellus with them (so Snape was able to know what they were planning), the former might supply fake information to Voldemort, about visiting the place at times and not finding anyone. And given the fact that Snape was the most trusted among the Death Eaters, Voldemort wouldn't question him further.
EDIT:
Also, don't overlook the fact that Snape might have told the secret only to Voldemort himself. Voldemort wouldn't have told Bellatrix or others about the location or have Snape reveal it to them, because there's no point of doing that; he was not the kind of chip chatting guy nor was any reason for them to know it. It was the abandoned headquarters of the OoP, nothing more. So, with only Voldemort and Snape knowing the secret, I can't think of the Dark Lord patrolling at 12 GP...
As for the "plot-hole" about Bellatrix knowing the location of 12 GP prior to the Fidelius Charm; she might knew that, yes, but the Fidelius Charm if placed on a building does not reveal the building itself. It has nothing to do with the address of it or the location (though the prior could be used by the Secret-Keeper to "add" a person to the inner circle of those who share the secret). Bellatrix could literally visit the street between 11 and 13 GP knowing that the headquarters are located at 12 GP and not see the building at all. Likewise, she could have told Voldemort about the address, or write it, or shout it but neither Voldemort nor anyone else could see 12 GP if they stood outside of the door. I imagine that something similar would have happened if she tried to Apparate there.
As Anduril_1251 said Voldemort and the Death Eaters did not know Snape was part of the Order and therefore would not know he knew of their safe house. Also, if Voldemort was better than Snape at Occlumency, Voldemort would have known Snape was playing him.
Snape was Dumbledore's as Harry explains to Voldemort in their last confrontation. Just as sure as Lily is dead, Snape would not have brought Harry to harm by revealing Grimmauld place to Voldemort and the Death Eaters.
That Snape was a member of the OoP, might have been known to Voldemort and the Death Eaters. But the fact that Dumbledore was the secret keeper, may not have been known to anyone. Hence, they may not have thought to ask Snape again for the secret of the location. Even if they did, Snape can get away by saying that Moody's curse has rendered him incapable of revealing the secret. He may instead have offered to go in and check the place on Voldemort's request, and reported (truthfully) the results, gaining his trust.
The point about Bellatrix not being able to reveal the secret, is explained by the points covered in this article about secret keepers.
Bellatrix knew the place before the Fidelius Charm was cast, but once it was cast she could not access it, as she was not told the secret explicitly by the secret keeper. Hence even after the death of Dumbledore, she was not a secret keeper, nor was she able to access it.
The above article also explains that the secret can not be tortured or forced or bewitched out of a secret keeper. They can only reveal it voluntarily (or inadvertently, as in the case of Hermione). Hence if Snape did not want to reveal the secret, none of the DE would be able to get it out of him, even by legilimency.
About Shell Cottage: Ron knew the location as he was told by Bill himself about it. I suspect that the charm was cast after Ron went to stay with them. Bill obviously revealed the secret to Ron, so he could go there in the future. Dobby took others to somewhere near the cottage, may be just outside (just as Death Eaters could apparate to just outside Grimmauld Place), and Bill and Fleur let them in, thereby revealing the secret to them.
Hermione became a secret keeper after Dumbledore's death, hence she was able to reveal its location to Yaxley by side-along apparition. Severus could not, being bound by Moody's curses.
Godric's Hollow: After the Potter's death, the original itself secret was not true anymore. Let's say the secret entrusted to Wormtail was that 'the Potters live at Godric's Hollow). After their death, this wasn't true any more. Hence the charm broke.
The charm may also break in case of change in ownership, to give access to the next owner. This makes sense, because you should not be able to cast a Fidelius Charm on somebody else's home who is unwilling!
There's a detail worth noting about the reveal of the Fidelius Charm: Dumbledore was able to reveal the location to Harry while not being there to reveal the location to Harry, by way of written note.
Which means that, before Sirius himself reveals that Dumbledore is the Secret Keeper to Harry Potter, Harry Potter didn't even know who the secret keeper was.
It's entirely possible that, given that the others in the Order of the Phoenix, and especially Sirius, didn't trust Snape enough to tell him - I don't have the text on hand to verify that this stayed the case, but it's entirely possible that Snape himself simply didn't know ahead of Dumbledore's death that Dumbledore was the secret keeper. Maybe Dumbledore himself revealed it to Snape, but...
We have to remember that Dumbledore did offer to be secret keeper to the Potters back in the day, and James considered it an insult to not trust his friends with his secret. Which probably would have worked, had it not been Peter Pettigrew whom he entrusted the secret too.
Voldemort and Snape both would know that the last time they had to deal with a Fidelius Charm, there was a presumption that Sirius was the Secret Keeper, even though it was actually Peter Pettigrew who was the Secret Keeper, and nobody had clearly attempted to test that theory (Since Peter Pettigrew was still alive, hypothetically, had the charm still been sort of up for others, revealing that Sirius couldn't reveal the location via truth serum or torture would prove he wasn't the Secret Keeper of the Potters).
Would Sirius Black also consider it an insult to trust Dumbledore as Secret Keeper over Remus Lupin, for example? After all, Sirius died earlier, and the Secret Keeper effect wasn't involved then (Presumably as Snape could reveal that he couldn't reveal the Secret of the Headquarters of the Order of the Phoenix then), so the Order of the Phoenix, while they entrusted Sirius to house them, clearly didn't entrust him to stay safe enough to be the Secret Keeper himself, and likely that he didn't trust himself with it. But as Snape found out in Prisoner of Azkaban, Remus still trusted Sirius enough to hear him out about Peter Pettigrew. And Sirius trusted Remus enough to re-use the Shrieking Shack as a hideout way past their original times there. More than he trusted Dumbledore at the time to listen to his story until he had literally no other choice, awaiting the Dementor's Kiss.
Snape had no way of verifying that, in fact, Dumbledore was the original Secret Keeper, short of trying to reveal the secret after he was dead.
The only time he would have absolute proof that the Secret Keeper was dead and the effect was in place was when Yaxley was able to be side-along Apparated there, at which point, even the trio wouldn't stay there, so it was kind of a moot point.
It was useless to go there. The order knew that Snape was with Voldemort and they knew it wasn't safe there. So they vacated the premises and took anything of importance with them. So you see, there was no reason for him to reveal the location and waste the Death Eater's time. I hope this answers the question.
its too much effort to find quotes, but i certainly remember sirius (or someone else, but i think it was sirius) explaining to harry why 12 GP was a good head quarters place. It had many protections of ancient magic outside of the fidelius charm that sirius' family had put into place. Maybe snape did tell of its location but the magic of the house somehow kept foes of its owner away? its not just guesswork as it is certainly mentioned the extra magic from dumbledore is just a precaution but the house has a lot of other powerful and ancient magic protecting it.
