According to Microsoft documentation you need to take extra steps to proxy the Public Folders to Exchange 2016 mailboxes.
In have simplified these steps, this is based on a single Exchange 2010 server and a single Exchange 2016 server. I understand this is applicable for co-existence with Office365 too.
You will need to change the name to match your Exchange 2010 server and also amend your AD name space instead of domain.local in the example.
On the Exchange 2010 server
New-MailboxDatabase -Server EX2010 -Name NewMDBforPFs -IsExcludedFromProvisioning $true
New-Mailbox -Name PFMailbox1 -Database NewMDBforPFs -UserPrincipalName PFMailbox1@domain.local -Password
Set-Mailbox -Identity PFMailbox1 -HiddenFromAddressListsEnabled $true
Set-MailboxDatabase NewMDBforPFs -RPCClientAccessServer EX2010
On the Exchange 2016 server
Set-OrganizationConfig -PublicFoldersEnabled Remote -RemotePublicFolderMailboxes PFMailbox1
Even after taking these steps Public Folders are not accessible. We can see that the DefaultPublicFolderMailbox is empty, you can see this using the following PoSh command.
Get-Mailbox | FL name, *public*
You can correct this for all users by executing the command below, make sure you change domain.local for you AD domain name.
Get-mailbox | Set-Mailbox -DefaultPublicFolderMailbox domain.local/UserAccounts/PFMailbox1
Now open Outlook 2016, open the folder view and the Public Folder list is there!