As a full-timer I can assure you that turning it off will save propane. The key is to plan your hot water usage ahead of time. In the Summer, I would only run my hot water heater for around 15 minutes before taking a shower. That's is for the day unless I needed to wash dishes that day. How can leaving it on all day be more efficient than that?
Also, if you haven't considered it, think about converting your HWH to a hybrid with one of the commonly available electric conversions. That's what I finally did, and now I have hot water heater 24/7 and it's on the park's dime.