Others / White Walkers
It only needs touch the skin of an Other (White Walker), but you better have time on your side if it is going to kill them...
And then he was stumbling forward, falling more than running, really, closing his eyes and shoving the dagger blindly out before him with both hands. He heard a crack, like the sound ice makes when it breaks beneath a man's foot, and then a screech so shrill and sharp that he went staggering backward with his hands over his muffled ears, and fell hard on his arse.
When he opened his eyes the Other's armor was running down its legs in rivulets as pale blue blood hissed and steamed around the black dragonglass dagger in its throat. It reached down with two bone-white hands to pull out the knife, but where its fingers touched the obsidian they smoked.
Sam rolled onto his side, eyes wide as the Other shrank and puddled, dissolving away. In twenty heartbeats its flesh was gone, swirling away in a fine white mist. Beneath were bones like milkglass, pale and shiny, and they were melting too. Finally only the dragonglass dagger remained, wreathed in steam as if it were alive and sweating. Grenn bent to scoop it up and flung it down again at once. "Mother, that's cold."
A Storm of Swords - Samwell I
We see the mere touching of the dagger causes the Other's skin to smoke, that while "dying" they melt/burn away, and it takes twenty heartbeats for all flesh to be gone.
So, honestly, if you are going to try and take down an Other, it would be best to stab it; otherwise you are going to be trying to press the dragonglass against its skin for quite a while and probably nowhere near as effective...
If you happen across a wight you would think that dragonglass would work as they comes from the same place.
The dragonglass dagger had melted the pale thing in the woods, true . . . but Grenn was talking like it would do the same to the wights. We don't know that, he thought. We don't know anything, really.
A Storm of Swords - Samwell II
But when Sam is attacked by a wight his dragonglass does nothing and later Melisandre confirms this.
Sam cleared his throat. "S-sire. The dagger . . . the dragonglass only shattered when I tried to stab a wight."
Melisandre smiled. "Necromancy animates these wights, yet they are still only dead flesh. Steel and fire will serve for them. The ones you call the Others are something more."
A Storm of Swords - Samwell V
However, Sam never actually hit the skin of the wight. He forgets that the blade was destroyed by the armor the dead man had been wearing.
There was no time to think or pray or be afraid. Samwell Tarly threw himself forward and plunged the dagger down into Small Paul's back. Half-turned, the wight never saw him coming. The raven gave a shriek and took to the air. "You're dead!" Sam screamed as he stabbed. "You're dead, you're dead." He stabbed and screamed, again and again, tearing huge rents in Paul's heavy black cloak. Shards of dragonglass flew everywhere as the blade shattered on the iron mail beneath the wool.
A Storm of Swords - Samwell III
In the end I would tend to agree with Melisandre though, but only because steel and fire will do better against the armor that the wights are wearing. While the Others have their own armor, the wights will be equipped with whatever the person who is turned was wearing when they died.