It's similiar to people saying you should leave lights on as the (re)turning them on uses more electricity and don't turn your car off if only stopping for a short time as the ignition will use up more fuel (an interesting thread on this on boards.ie or octane.ie a short time back. I think they got a figure from somewhere giving 3 1/2 minutes for an average family car to be stopped before it should be [from a fuel economy point of view] turned off - but I am a little hazy on the details).
An emersion is very expensive. Obviously each home is different, but I'd imagine for leaving it on all the time to be more efficient you'd need to be turning it on numerous seperate times (morning, mid-day, evening and night). The best way to save with the emersion bill would be to organise things so it only needs to go on once a day, reduce the amount of water used where possible (good for the environment too) and don't forget to turn it off.
I've never seen any stats to back up leaving it off, but I've never seen any to contradict it either!