Severus Snape had a painful death

Why is Dumbledore asking Snape in HBP?

The other answers here are not wrong. But they do not contain the (whole) correct answer. Namely:

Dumbledore was afraid of Greyback.

This is what causes Dumbledore to ask Snape. He knew he was going to die one way or another. He had accepted death and was not afraid of it. But he feared that he might die from another Death Eater (like Greyback), which would cause his death to be protracted and traumatic.

"You alone know whether helping an old man will harm your soul To avoid pain and humiliation "Dumbledore said." I beg you this great favor, Severus, for death will come as sure to me as the Chudley Cannons will come last in this year's league. I admit I should quick, painless exit over the lengthy and chaotic affair, for example when Greyback is involved - I hear that Voldemort has recruited him? Or the dear Bellatrix, who likes to play with her food before she eats it. "
His tone was bright, but his blue eyes pierced Snape, as they had often pierced Harry, as if the soul they were arguing about was visible to him. Finally Snape nodded briefly again.
(Deathly Hallows, Chapter 33, The Prince's Tale) .

Note that Dumbledore wants to avoid pain and humiliation and is particularly concerned about Greyback and Bellatrix. When he's disarmed on the Astronomy Tower, Dumbledore knows his time must be up. Either Snape will kill him or one of the other Death Eaters will. Bellatrix isn't there to do it, but as he feared it's Greyback.

"Is that you, Fenrir?" asked Dumbledore.
"That's right," croaked the other. "Nice to see Dumbledore?"
"No, i can't say i ... "
Fenrir Greyback grinned and showed sharp teeth. Blood ran down his chin and he licked his lips slowly and obscenely.
"But you know how much I like children, Dumbledore."
"Am I supposed to assume that you are attacking without a full moon? That is most unusual ... You have developed a taste for human flesh that cannot be satisfied once a month?"
"That's right," said Greyback. "Shocked you, doesn't it, Dumbledore? Scares you "
"Well, I can't pretend it doesn't put me off a little," said Dumbledore.
[...]
And he lifted a yellow fingernail and grabbed his front teeth and stared at Dumbledore.
"I could do you for later, Dumbledore ..."
(Half-Blood Prince, chapter 27, The Lightning Struck Tower) .

Dumbledore was brave enough to stand up to Greyback. He wouldn't openly reveal his fear in front of the Death Eaters. But that doesn't mean he wasn't scared. He is unarmed and defenseless. Greyback demonstrates his very real desire to tear him to pieces. It's not a pretty death, as Dumbledore himself says in the earlier quote. Although he was also interested in saving Draco's soul, as Dave Johnson says, this is really what he's talking about out of self-interest . Snape is his only way out of a gruesome, painful death. But Snape wasn't ready to kill Dumbledore - maybe out of love / respect for Dumbledore and maybe because killing someone is just a big deal (Snape has to be willing to do Dumbledore's request). So Dumbledore asks him to keep his word and finish him off - he knows the fate that awaits him should Snape fail.

Möoz

Good point, besides, Greyback could have thrown a wrench in the works by not killing Dumbledore but turning him over.

The dark lord

@Mooz if he could be rotated he would become a powerful ally ... probably more powerful than luke tbh. Dumblewolf is a scary notion. Even though the curse wouldn't have killed him anyway?

Bellatrix

Ooo I scare Dumbledore!

Pryftan

"I know the fate that awaits him should Snape fail." And the fate of the school and the world because Severus was not the most respected of Voldemort. I also think this has something to do with what Hagrid tells Harry: hearing Snape and Dumbledore arguing because Snape no longer wanted to do [which he had agreed].