Is that possible to prove that 'I do not know something' (i.e, the private key for a Bitcoin wallet) using cryptography? Is that possible to prove that 'I do not know something' (i.e, the private key for a cryptocurrency wallet) using cryptography?

The problem might be nonsense at first glance. But the intuition for this might be to prove that I do not hold the crypto coins under a specific wallet.
You are asking for some cryptographic function whose output changes based on whether or not your mind holds a particular piece of information. There is no function which is capable of knowing what is in your mind. You can cryptographically prove ownership, but you cannot do the opposite.