I have a 40 gallon water heater in my summer home. Yes it does get very cold. I want to shut off my water and the gas furnace, but leave the water heater on "vacation". Will it be okay without draining?

I trust you will at least drain all the pipes. Hot water heater should be OK.

If you don't drain all the pipes, including toilets and drain traps you must keep heat in the house. If you are going to drain the pipes, you may as well drain the water heater also and turn it off.
If you are going to keep heat in the house you can just turn the water heater off. No point in using the electricity of gas to heat the water, even at the vacation setting.

Regardless, you should turn the water to the house off.

How cold will it get inside this home during the winter? It is a waste to heat a water heater even on vacation mode for the entire winter, drain it and the pipes.