Customer needs to be hosting the inbox in Exchange Online to be able to setup a Azure Application. Exchange Online is always a part of Office 365, but can also be configured for on-premise Exchange Servers in so called Exchange Server hybrid deployments. Note that the inbox that Lime-Inbox uses must be hosted online. What versions of on-premise Exchange that supports hybrid deployment can be found here: Hybrid deployment prerequistites.
Customer needs to grant permission for Lime to use their domain if autoreply is used. Instructions can be found here.
TRAML is used for auto replies. This is only applicable if an auto-reply is needed. See TRAML Requirements.
Lime needs access to internet (to Microsoft Graph API).
Public access to Lime is needed (or whitelisting of messages from the Azure AD Application to the lime server / relevant endpoints)
The customer needs to make sure that the external DNS that is configured for the inbox and recovery endpoints is also reachable from within the application-server. Otherwise the recovery functionality won't work.
More detailed information regarding networking access can be found here.