It's possible, but we actually have different uses for this phrase. I mean, we'll understand you but we'll also realise that it's not a natural question in the context.
The natural question you want is, "Is there anything that you don't do?"
"What do you never do?" is usually part of a longer question, and suggests the worst possible option in the situation. It basically means, "What must you avoid doing at all costs?" The "you" is impersonal.
For example, "What do you never do with a flour sieve?" (internet example) The response is, "You never..."