IMAP is a client access protocol. It only works on one particular server, and it presents the same folder structure and contents to all users connected to that server and the same user account. IMAP cannot be used to synchronize folders and content within folders from one server to another.
Outlook is a groupware client. It can use IMAP to connect to one server per configured mail account, but it cannot synchronize the contents of multiple mail accounts (you probably don't want that anyway, it sound more like you want to synchronize the folder contents for one mail account across multiple servers.
AFAIK the synchronization of folders and contents will only work if all servers that participate in the synchronization are of the same make/model, i.e. they are all MS Exchange servers or they are all parts of a Zimbra cluster or similar.
In consequence I don't think what you have in mind is actually doable.