I'd shut it off, personally. Even if it cuts on for a bit mid day just to keep things hot, that's still wasted propane if I'm not about to need hot water.
Rene T said:
I have gas/electric and leave mine on electric all the time even if were gone all day. I do shut the water off just in case..
Rene
I'll suggest that you may want to reconsider this. general "best practice" rule of thumb I've been taught about working on residential water heaters is that if you shut off the water be sure to shut off the power.
The thought is this.... & I'll just make up a scenario....
let's say someone came up to your RV while you were away and tried to use the outdoor shower. Also, you happened to leave one of your sinks dripping, just a bit..... that dripping faucet might be enough to let air into the hot water pipe which might allow the water in the heater to siphon out. Now you have an empty water heater sensing cold so it turns on.... kills the element...or worse.
I bought a foreclosure house that was less than 1 year old when the owner foreclosed and moved out. Two water heaters. It had been vacant for a long time. the plumbing system had been winterized. I don't remember for sure but I think the power was on. If not, I'm pretty sure it had been at some point.
Anyway, when i bought it I had to replace 4 burned out elements
edit: I remembered that the power was on, because we had to run the pool pump before my lender would process the sale...