Hi Andy,
I suggest you deactivate the Office ProPlus on the device and reactivate it again with the new user’s account. No matter the device is a Windows PC or a Mac.
If we just remove the account from Office applications, it may have some problems in the future since it doesn’t equal to deactivate.
You can follow the steps below to complete this:
1. Login to Office 365 admin portal with the old user’s account (only this account has the permission to deactivate the Office 365 ProPlus he had used before)
2. Click the Settings menu 
on the top right corner of the page and click Office 365 settings.
3. Click software and in the coming page, you will see which computers the old user has installed Office.
4. Click deactivate for the appropriate device. After deactivating, the Office 365 ProPlus on that device will go into reduced functionality mode. Once you open the Office applications, you will see the popup window to let you reactivate it with the new user’s account.
If there are any questions, please feel free to post back. Meanwhile, you can tell me the subscription of Office 365 you are using.
Reference:
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http://technet.microsoft.com/en-us/library/gg982959.aspx#BKMK_ReducedFunctionalityMode
Thanks,
Elaine Hao
