When someone asks a question, is the reply, "That I don't know" correct? Or is the simpler "I don't know" a more correct or proper response?
Normally you just say:
I don’t know.
Sometimes when you want to draw attention to the thing unknown, you might venture to add an object:
I don’t know that.
But it would be rare indeed to apply inversion to put the now-important bit first:
That, I don’t know.
I disagree with tchrist on "That I don't know" being unusual. It is very common but has a different connotation. Saying "That I don't know" would be said when you had already answered a question or two, but did not know the answer to the current question. As in
What time are we leaving?
Is Barry going with us?
That I don't know.