It seems that Dumbledore is the only person in the staff who knows Quirrell is dead.
The account of Harry, Hermione, and Ron's efforts to stop Quirrell from getting the stone was widely propagated, probably by the latter two.
According to Professor Dumbledore:
“What happened down in the dungeons between you and Professor Quirrell is a complete secret, so, naturally, the whole school knows.”
Harry Potter and the Sorcerer's Stone
If the “whole school”was aware of these events, the staff could hardly have been ignorant.
Further, a number of faculty, such as McGonagall and Snape, were members of the Order of the Phoenix, and thus Dumbledore would likely have apprised them of anything Voldemort-related.
Even if we assume that there was no announcement of Quirrel's death at the time, we know that at least a few members of the staff found out about it at some point.
Professor Slughorn knows that Quirrel died, as Harry explicitly tells him in the beginning of Half-Blood Prince:
"Most of the teachers aren't in it, and none of them has ever been killed — well, unless you count Quirrel, and he got what he deserved seeing as he was working with Voldemort."
And the fact that Slughorn didn't respond to this may even indicate that he already knew that Quirrel had died before Harry told him.
We also know that Voldemort told the Death Eaters in the graveyard in Goblet of Fire that the teacher he had been possessing had died:
"Then... four years ago... the means for my return seemed assured.
"Oh, he seemed the very chance I had been dreaming of... for he was a teacher at Dumbledore's school...
"The servant died when I left his body, and I was left as weak as I had ever been," Voldemort continued.
While Voldemort never mentioned Quirrel's name, he told them the year, and that the teacher had died, so it would be pretty simple for any of the Death Eaters present to figure out that it was Quirrel who died. Now, there were several Death Eaters on the Hogwarts staff over the years. Barty Crouch Jr. taught in Hogwarts in Goblet of Fire; however, he was not present in the graveyard so he had not heard Voldemort's speech there. It is possible that Voldemort told him the same information on another occasion, but that does not appear in the books. Similarly, Severus Snape taught at Hogwarts throughout the entire series, yet he too was not present at the graveyard speech. Again, it is possible that Voldemort told him separately, but it's not mentioned in the books. In Deathly Hallows, Alecto and Amycus Carrow taught at Hogwarts. They were not specifically mentioned in the graveyard, but there is no reason to assume that they weren't there. Thus, the two Carrows would be members of the staff that probably knew that Quirrel was dead.