BizTalk does store messages. Messages associated with active service instances (i.e. during processing) are stored in the MessageBox, but you can also enable tracking of full messages received and sent to be stored in the tracking databases.
However, it does not sound to me like this is what you really want. Comparing the entire message contents to check for duplicate messages would be, at the very least, extremely expensive and really unnecessary most of the time. If your messages have some kind of natural identifier (i.e. they contain a message-id field or something else that identifies the request uniquely) then what I'd do is just to store that ID in a DB (not necessarily the entire message content, just the ID) and then compare that during the receive (either in a pipeline component for maximum efficiency, or at the beginning of an orchestration).