From Unofficial BOINC Wiki
The good news is that Off-Line Processing means that work is being done, so we don't have a problem with equipment being Off-Line, but the equipment that is being used to do the processing is not connected to any of the systems that are processing external requests. In the world of Distributed Computing we have equipment that is available and processing requests from the Participants such as the Project's Web Site, Project's Data Server, Project's Scheduling Server which means of course that all of this equipment is "On-Line".
However, there may be requirements to process a long running query against the BOINC Database and this request might impact the Project's Data Server ability to respond to the requests of the Participants if it was run on that Server. In this case, a copy of the Database might be made to another Server that is not processing external requests, and then this long-running query would be processed on that Server. In this case, the query is run on a server that is "Off-Line".