What is the latest mac os that works with QuickBooks 2016

That would be Mojave (10.14.x).

For Catalina (10.15.x) you must upgrade to QB 2019 or 2020.

QB 2022 requires BIg Sur (macOS 11.3) or later.

QB 2023 requires Monterey (macOS 12.x) or later.

Best ask at the QuickBooks site. There's an online agent there you can chat with, who might help.