Final answer:
Public folders require an on-premises Exchange server, while shared mailboxes, email encryption, and mobile device management are supported in Exchange Online.
Step-by-step explanation:
The feature that requires an on-premises Exchange server is Public folders. Exchange Online does not support public folders, so if your company relies heavily on this feature, it should be taken into consideration before migrating to Exchange Online.
Shared mailboxes, email encryption, and mobile device management are all supported in Exchange Online and do not require an on-premises Exchange server.