It's possible that you may not be able to delete a contact in Outlook due to an issue with your account settings. To troubleshoot this, I suggest following these steps:
1. Check the permissions of the folder containing the contact. Make sure it allows users to modify or delete contacts.
2. Go into File > Options > Advanced and make sure ‘Ignore other applications’ is unchecked under the Send/Receive heading.
3. If you are using a shared mailbox, check if you have sufficient privileges to delete contacts from it; otherwise, ask your system administrator for assistance with changing permissions on your account so that it can be done successfully.
4. Try deleting the contact in OWA (Outlook Web Access) instead of Outlook desktop client as this might help resolve any potential issues caused by local profile corruption or incorrect synchronized data between clients and servers - simply navigate to https://outlook[dot]office365[dot]com, log in using your Microsoft credentials, select Contacts from within Mail section and find & remove/delete desired one(s).
5. Finally try resetting Office 365 Outlook cache - open CMD prompt as admin and type 'ipconfig /flushdns' followed by 'ipconfig /registerdns', then restart Outlook application again . This should do the trick!