In Star Wars Episode VI: Return of the Jedi, when Luke was being tortured by the Emperor's Force Lightning, why didn't Vader just go pick up Luke's lightsaber (which Luke had previously thrown aside) and just stab the Emperor from behind?
There are two reasons Vader didn't "just" pick up Luke's lightsaber:
Luke threw it away.
In the movie, Luke stands on a walkway. (the one whose railing his lightsaber cuts, when Vader leans on the railing and Luke cuts off his hand).
He throws away his lightsaber when declaring "he's a Jedi, like his father", and it is possible that the Lightsaber falls off down the chasm over the walkway (though disputable - the angles may be wrong, and neither novelization nor the script mention it falling although they explicitly mentioned that Vader's fell. According to comment by @RedCaio, it did NOT fall down as Luke wore it on Endor at the movie's end) .
The main point (no matter where it fell) is that I have just re-watched that entire Death Star scene, and I don't see Luke's lightsaber lying anywhere near him on the floor, at any time between him throwing it away, and him picking near-dead Vader up and carrying someplace else.
In ROTJ Novelization (EPISODE VI; STAR WARS®: Return of the Jedi® by James Kahn), it simply says:
He hurled his lightsaber away. 'Never! Never will I turn to the dark side! You have failed, Palpatine. I am a Jedi, as my father was before me.'
Hurled also implies that it flew off some distance. Doesn't say it fell down (unlike Vader's) but didn't just fall right near where Luke was standing.
Interestingly, neither the movie nor the novelization ever again mentions Luke's or Vader's lightsabers again. (I got dibs on asking how he retrieved one/both)
And it's clear that Vader was 100% NOT up to the task of Force-summoning the lightsaber, as the novelization shows in detail:
At that instant, Vader sprang up and grabbed the Emperor from behind, pinning Palpatine's upper arras to his torso. Weaker than he'd ever been, Vader had lain still these last few minutes, focusing his every fiber of being on this one, concentrated act - the only action possible; his last, if he failed. Ignoring pain, ignoring his shame and his weaknesses, ignoring the bone-crushing noise in his head, he focused solely and sightlessly on his will - his will to defeat the evil embodied in the Emperor.
Palpatine struggled in the grip of Vader's unfeeling embrace, his hands still shooting bolts of malign energy out in all directions. In his wild flailing, the lightning ripped across the room, tearing into Vader. The Dark Lord fell again, electric currents crackling down his helmet, over his cape, into his heart.
Vader stumbled with his load to the middle of the bridge over the black chasm leading to the power core. He held the wailing despot high over his head, and with a final spasm of strength, hurled him into the abyss.
Note that the novelization confirms that this was "the only action possible" - which, given that Lightsaber stabbing is an obvious action to occur to both Lucas out of Universe AND to Jedi-turned-Sith Vader in-universe, means it was somehow impossible for him to get Luke's lightsaber.
To answer Sachin's comment, Vader's lightsaber is also not there - it fell off together with Vader's cut-off wrist and also fell down from the walkway. From the Screenplay:
Vader's sword clatters uselessly away, over the edge of the platform and into the bottomless shaft below.
Here's all the screenshots of the floor with no lightsaber around their location where Vader could easily pick it up: