Mining a transaction with a 0 BTC output
Do you want to create p2pkh output and send zero satoshies on it? Such output is non-standard (dust) and will not be relayed and mined (but it is possible)

Or may be you want to create nulldummy (OP_RETURN) output? This is possible

BTW, sometimes miners confirm non-standard transactions, for example https://blockchain.info/tx/b4ea693a9d2d7e32b66777964fce13da302bed024bd9fd4651919da650fb1c40 (it is p2pk output)