Restrict Editing of Office 365 Profile information

You have to make changes in Office 365's Remote Powershell. Let me know if you don't know how to do so. https://eightwone.com/2011/03/31/disabling-editing-account-information-in-owa/ gives some guidance on limiting user access to various functions, but it can get fairly complicated if you go beyond just removing access to modifying contact information (for instance, if there are options that you want to block that don't fall under the MyContactInfo role assignment, you'll have to be more granular). Granting HR access to Contact Information modification for other users is also a little tricky. It all involves making changes to the Role based access control system in Office 365 and modifying role assignment policies. You would basically need to create a group for HR, create a role, create a role assignment policy that allows users to modify contact info for all users (This usually involves adding Powershell cmdlets and options that the group is allowed to use), assign the policy to the role, then assign the role to the HR group. It's a huge subject, really: https://technet.microsoft.com/en-us/library/jj200692(v=exchg.150).aspx explains more