Which is correct?
You are an idiot who can’t write.
You are an idiot that can’t write.
(I’m serious. Flames will be ignored.)
When all else fails, ask Cecil.
You are an idiot who can’t write.
The “who” refers to “you”, a person.
You are an idiot which can’t write? (Ducking) 
Billdo is correct. However, it would also be:
You are the idiot that can’t write.
I think.
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Shadow of the Pigeon -
Weirdo of the Night
Man, you guys are mean! Jab1 just asked a simple question. Even though he said he’d ignore flames, that’s no reason to take him up on it.
Wrong thinking is punished, right thinking is just as swiftly rewarded. You’ll find it an effective combination.
You are an idiot. Who can’t write?
I say either one.
As long as the message gets through.
What difference does it make? The guy’s an idiot. He won’t know the difference.
Oh, boy, is Rousseau gonna be pissed. Check this Pit thread. The writer in question is AvenueB-dude.
When all else fails, ask Cecil.
I meant to say, “What difference does it make? If the guy’s an idiot, he won’t know the difference.” An edit feature would be nice sometimes. 
BTW, I checked with my Office 97 and it says that either “You are an idiot that cannot write.” or “You are an idiot who cannot write.” are grammatically correct. It doesn’t like “can’t” though. I am an idiot that can’t tell the difference.
I’m pretty sure you should use “that” to distinguish between one or more objects (e.g. “You are THE idiot that can’t write” as stated above), so if anything, it’s a question of:
“You are an idiot which can’t write.”
vs.
“You are an idiot who can’t write.”
…and of course at that point you’re making a judgment call about the idiot’s fitness for humanity.