I doubt it as the water is what? 180F? Even if the coils were hanging in the empty tank for days with hot water in them, doubt they would fail. Just not hot enough.
BTW, most Class A RV's I'm familiar with the water system can withstand 20F weather for many days before you need worry. You didn't tell us the year or brand of RV you have but newer RVs, say from 2000 on, also have PEX plumbing that is much more tolerant of freezing then the older grey stuff.
So partially fill the onboard water tank at the first overnight stop you stay at, than fill your water heater tank...your system will be fine. Just do the regular...before bed, turn off the shore water, or turn off the pump if you're boondocking.