The stake technology specialist has primary responsibility for maintaining the computers in the FHC, so he definitely needs administrative access to all the computers.
Beyond that, I don't see any specific policy, except that the
Policy and Guidelines For Computers in Family History Centers says that the STS "Takes direction from the stake presidency regarding the management of Church computers in family history centers."
In our stake, I (the STS) am the only one with Admin access to computers. The usernames and passwords used for that access are in a confidential document that the stake president has (in case I am somehow incapacitated). In our case, the FHC director is not particularly technical, so there would be no reason for her to have admin access anyway.
I did find a
support article 103751 that says that when a new FHC director is called, the previous director will "Provide you with the Administrator passwords for the computers."
Questions that can benefit the larger community should be asked in a public forum, not a private message.