I would like to remove Posix from the Group Policy Object Computer Configuration\Policies\Windows Settings\Security Settings\Local Policies\Security Options\System settings: Optional Subsystems.
Do the GnuWin32 tools require the POSIX subsystem to run?
No. The GnuWin32 tools are (as the name implies) Win32 applications, i.e., they use the Win32 subsystem, not the POSIX subsystem.
Of course, no matter what advice you may receive from strangers on the internet, you should test all your critical applications/services on a test system before making this change on any production systems. :-)
I've never installed/enabled it and GnuWin32 works fine. I don't think the two are related in any way.