One of the items in Odin's Vault survived, namely the Tesseract: we see Loki pause on it in Thor: Ragnarok and it is confirmed in Avengers: Infinity War that he did indeed take it.
Thanos: The Tesseract, or your brother's head. I assume you have a preference.
Loki: Oh, I do. Kill away! ALL RIGHT, STOP!
Thor: We don't have the Tesseract. It was destroyed on Asgard.
Loki glances at Thor like he knows something he doesn't. He lifts his right hand into the air and the Tesseract reveals itself.
Thor: You really are the worst brother.
Avengers: Infinity War
The above dialogue is quite telling in that if Thor felt Ragnarok destroyed the Tesseract he likely felt it destroyed all of the other items as well. And how could it not, really? Asgard itself just disintegrates:
I just want to specifically mention two other items here: the Eternal Flame and the Infinity Gauntlet.
The Eternal Flame is a flame so not likely to be able to get destroyed; it's also possible it merged with Surtur at the the end of Thor: Ragnarok but that's unclear. If the latter it could have survived if Surtur survived but again that too is unclear.
The Infinity Gauntlet in Odin's Vault is a fake, Hela calls it out as such.
On that last point Hela mentions most of the stuff in there is fake.
SKURGE follows her. Hela walks towards the Infinity Gauntlet.
Hela KNOCKS the gauntlet off its pedestal.
Hela: Most of the stuff in here is fake.
Next: The Casket of Ancient Winter.
Hela: Or weak.
A lot of it being fake means it doesn't really matter if it was destroyed or not. The reasoning for the Infinity Gauntlet being fake has been cleared up by Kevin Feige and it's essentially because they were trying to cover up for the Easter Eggs in Thor. However, he also gives the in universe answer of Odin doing it to look more of a protector than he is.
That came about because in Thor 1, the reason that it's in this movie is because someone who went through that movie, frame by frame, looking for Easter eggs was like 'Oh! The Infinity Gauntlet!' And then from that point on, they were like, 'That damn Infinity Gauntlet, what are we going to do with that thing?' It's like, there, but it also [exists] somewhere else in the universe. How do we deal with that? And then we realized like, Odin... Hela goes through and she goes, 'Fake. All of that stuff that's in here is fake.' Basically the thing we were trying to go for was that Odin somewhere along the line realized that everyone knew that the Infinity Gauntlet was on Asgard and in the safe keeping of Odin, then no one would come looking for it. So he made a fake one, and let everyone believe... Basically, a fake launch code.
CinemaBlend, Why Thor: Ragnarok Included That Scene With The Infinity Gauntlet
The /Film interview with Kevin Feige goes into more depth on the matter if you want to read it.
So, we have fake and weak items in Odin's Vault at the time of Ragnarok with the decent one, Tesseract, being saved by Loki. It seems reasonable to conclude that most were destroyed, especially so if they are fake.