Imagine if an unprivileged user (i.e. not a member of local administrators) found an NTLM hash of a user within the local administrators group. Could the unprivileged user obtain admin privileges? TLDR; Yes!
At first I thought maybe you could Pass-The-Hash to local services like WMI, SMB, etc using something like https://github.com/Kevin-Robertson/Invoke-TheHash. I quickly discovered that those services do not allow user credentials to be specified for local connections. If anyone has a workaround to this, without requiring a second machine on the network, please let me know @aceb0nd.
The PTH technique could work from a remote system to the compromised device, depending on LocalAccountTokenFilterPolicy and FilterAdministratorToken, but I considered that cheating.
I then realised, the Windows change password functionality only requires knowing the users NTLM hash. I used Mimikatz to update the password of the administrative account.
lsadump::changentlm to update the administrative account password
I then used runas to execute PowerShell as the admin user.
runas to execute a program with administrative privileges
A UAC bypass would be required to further escalate privileges past the filtered admin token but I consider that out of scope for the question. In conclusion, if an unprivileged user finds an administrative NTLM hash, they can compromise the account fairly easily.