You are remembering "Go Starless in the Night" by Roger Zelazny. One of many stories reprinted within the Zelazny collection Unicorn Variations.
The plot is much as you describe. The viewpoint character is Ernest Dawkins, a man from the late Twentieth Century who was slowly dying (cancer would be my guess, but I don't think the text specifies the details), and so he arranged to be cryogenically frozen, on the theory that someday scientists might find a way to safely defrost his body and also heal his ailment to make him good as new.
It appears that Dawkins has stayed frozen (and asleep) for a long, long time, until just now, when he starts hearing voices who want to converse with him.
They speak English . . . but with odd grammar and syntax at times, suggesting that either the language has changed noticeably over several generations, or else it is not their native tongue, and they're just improvising. The voice that does most of the talking (although Dawkins thinks he hears others in the background) claims to be conducting archaeological research, and says that there seems to have been a terrible war . . . so messy, and so long ago, that it's very hard to say who "won." (If anyone did?)
Here's an excerpt in which the protagonist is told just how this conversation is even possible. It comes right after a voice has declined to discuss whether or not a modified form of English is its native tongue. First speaker in this passage is Dawkins; second voice is his mysterious interrogator.
"As you would. Then will you tell me about my situation? I am more
than a little concerned. I can't see or feel anything."
"We are aware of this. It is unfortunate, but there is no point in
misrepresenting to you. The time has not yet come for your full
arouse."
"I do not understand. Do you mean that there is no cure for my
condition yet?"
"We mean that there is no means of thawing you without doing great
damage."
"Then how is it that we are conversing?"
"We have lowered your temperature even more -- near to the zero
absolute. Your nervous system has become superconductor. We have laid
induction field upon your brain and initiated small currents within.
Third space, left side head and those movement areas for talk are
now serving to activate mechanical speaker here beside we. We address
you direct in the side of brain places for hearing talk."
One bit that your memory may have blurred -- it doesn't seem as if the interrogator already knew that Dawkins had worked with chemical and biological weapons on behalf of his government. They simply had found him (and perhaps other frozen "survivors" of the war? We aren't told) in a self-contained nuclear-powered storage facility, and rigged up a way to ask him questions about general conditions during his lifetime.
But when Dawkins mentions that his work involved knowledge of three large caches of chemical and biological weapons stashed away for a rainy day, the interrogator starts asking about where those things were kept. The excuse, of course, is that the interrogator's culture is extremely peace-loving and wants to make sure that there will be no nasty accidents if someone happens to stumble into the wrong place.
Dawkins, as you recalled, realizes that he has no way of independently confirming any claims made by unknown entities living in an unknown future. He doesn't even know if he's being questioned by fellow humans. When he explains his doubts, the interrogator soon turns to threats of the nasty things they can do to him if he won't cooperate, and he then feels validated in his decision not to give them any data regarding hidden Weapons of Mass Destruction.
As you said, they finally leave him "switched on," but without any further input reaching his brain. The equivalent of being left in a modern "sensory deprivation tank" indefinitely. He's mentally retreating into a fantasy world as the story ends. (And we still don't know if the voices belonged to humans, or alien conquerors, or what.)
