Why didn't Dumbledore tell the Order about Horcruxes?

Question

We know that Harry didn't have to destroy the Horcruxes himself:

"Ron - ?"
The sword flashed, plunged: Harry threw himself out of the way, there was a clang of metal and a long, drawn-out scream. Harry whirled around, slipping in the snow, wand held ready to defend himself: but there was nothing to fight.
_{-Harry Potter and the Deathly Hallows, Chapter 19 ("The Silver Doe"), by J.K. Rowling}

So why didn't Dumbledore tell any of the Order about the Horcruxes? Did he think it would be too dangerous? Did he not trust them?

Answer 1

He didn’t want the Dark Lord to know he knew, and secure them.

The reason that Dumbledore keeps his knowledge about the Horcruxes secret, and presumably this is the same reason he didn’t tell the Order, is because he didn’t want word to get around that he knew so many of the Dark Lord’s secrets. This is likely so the Dark Lord would stay confident his Horcruxes would be safe, and he wouldn’t decide to secure them better or change their locations.

“Sir, am I allowed to tell Ron and Hermione everything you’ve told me?’

Dumbledore considered him for a moment, then said, ‘Yes, I think Mr Weasley and Miss Granger have proved themselves trustworthy. But, Harry, I am going to ask you to ask them not to repeat any of this to anybody else. It would not be a good idea if word got around how much I know, or suspect, about Lord Voldemort’s secrets.”
_{^{- Harry Potter and the Half-Blood Prince, Chapter 10 (The House of Gaunt)}}

This was sound reasoning - as soon as the Dark Lord finds out Hufflepuff’s cup was stolen, he decides to check on them all and redouble protection around them.

“But to be sure, to be utterly sure, he must return to each of his hiding places, he must redouble protection around each of his Horcruxes …”
_{^{- Harry Potter and the Deathly Hallows, Chapter 27 (The Final Hiding Place)}}

By not telling anyone, including the Order, Dumbledore ensured that the Dark Lord remained confident in his defenses for as long as possible - by the time he began checking on them, the Dark Lord only had two left, the diadem and Nagini. The diadem was destroyed while he was still in the process of checking on them, so he didn’t have time to move it either. Dumbledore prioritized keeping the Dark Lord from finding out he knew. Though the Order should be theoretically trustworthy, last time the Potters died because of a spy in their ranks.

Answer 2

It appears that Dumbledore was afraid of Voldemort finding out that he knew about the Horcruxes. As we find in a conversation in Half-Blood Prince (my emphasis):

He turned to go, then another question occurred to him, and he turned back again. “Sir, am I allowed to tell Ron and Hermione everything you’ve told me?”

Dumbledore considered him for a moment, then said, “Yes, I think Mr. Weasley and Miss Granger have proved themselves trustworthy. But Harry, I am going to ask you to ask them not to repeat any of this to anybody else. It would not be a good idea if word got around how much I know, or suspect, about Lord Voldemort’s secrets.”

“No, sir, I’ll make sure it’s just Ron and Hermione. Good night.”

Additionally, at the end of Deathly Hallow Harry realized that Dumbledore also didn't want to waste anyone else's life. Harry had to die anyway, so Dumbledore decided to use him as the Horcrux hunter:

Dumbledore’s betrayal was almost nothing. Of course there had been a bigger plan; Harry had simply been too foolish to see it, he realized that now. He had never questioned his own assumption that Dumbledore wanted him alive. Now he saw that his life span had always been determined by how long it took to eliminate all the Horcruxes. Dumbledore had passed the job of destroying them to him, and obediently he had continued to chip away at the bonds tying not only Voldemort, but himself, to life! How neat, how elegant, not to waste any more lives, but to give the dangerous task to the boy who had already been marked for slaughter, and whose death would not be a calamity, but another blow against Voldemort.