Usable Li-Ion batteries don't self-discharge at these sorts of rates and self-discharge wouldn't appear to explain why the phone only discharges rapidly when powered off.
My phone has the same problem, though it is intermittent. I turn my phone off at night and back on the next day, though sometimes the phone remains off for up to 3 days. The phone will always restart at 100% charge (I always charge it before shutting down), regardless of how long it's been off. However, about once a month the battery will be completely empty the next morning (less than 12 hours after recharge/shutdown), which means that I'll start that day with a dead phone. If the phone is left powered on instead, it will lose only about 4% overnight. I'm now leaving my phone turned ON overnight just to avoid draining the battery, which is just nuts! It's hard to imagine why a faulty or aging lithium-ion battery would behave this way.
Just to make sure I've been clear, the battery drains rapidly only when the phone is turned OFF and in my case it only does this intermittently. @whypham seems to be experiencing it every time he/she turns off their phone.