It doesn't turn itself off, but leaving it on constantly emits a signal that the bike searches for. Eventually the battery in the fob will fail and the bike won't be able to find it. It should still function as the spare does though in the emergency start procedure.
A friend of mine collected a new Storm GT last year, and I went with him. On the handover, the sales woman had no idea what the spare keys did, or what the emergency start procedure was. I had to show them. Take the battery out of the fob, and try the emergency start procedure with the "dead" fob ? It would of course mean that you don't have to carry a spare key for emergency use ?