The main savings by closing the BBC Three linear channel will be a massive cut in the commissioning budget. Put simply, they will spend less as they will buy/make fewer programmes as they won't have a 7 day a week linear schedule to fill. As others have said, we expect BBC Three branded long-form content to be available on-demand i.e. via iPlayer. There will also be short-form content for Facebook, YouTube etc.
This is the main saving - a massive cut in the commissioning budget. The long-form shows commissioned for BBC Three's on-demand service will also be broadcast on BBC One or BBC Two as well. There will also be a smaller saving in not having to fund a channel operation. There won't need to be a playout director, the associated support teams, and there won't need to be detailed scheduling for the service (though scheduling of the publication of on-demand content will be needed). The BBC One+1 service will presumably just be a straight time-shift of BBC One HD England in England, with a different holding caption for the 2225 regional opt (as I doubt they will time shift every English region on SD?)
There won't be a capacity saving as the the BBC Three / Three HD mux space being freed up will be re-allocated to BBC One+1. (Unless they decide not to offer a BBC One+1 HD and sell the time-exclusive spectrum to a third party - but I doubt that will happen)
One thing that hasn't been suggested AFAIK, but which would be an interesting approach, is a curated on-demand playlist for the on-demand service which you can watch as a sort of channel that you can jump around in.
This is what SVT Flow does in Sweden
http://www.svtflow.se Effectively it is an on-demand service, but there is a playlist of on-demand stuff which is only a short step away from being a linear channel, but without trails, continuity etc. Much cheaper to schedule and operate and you don't have to fill a fixed amount of time. You can start viewing a show at any time, and when it finishes the next scheduled show plays. But you can jump to the next show if you want.