Will draining a water heater damage it?
Will shutting down my main water line inlet and draining my lines in
preparation for a freeze damage my gas hot water heaters?

Is it necessary?, we do  not know about your situation but to drain your whole system and shut it down the following is a general overview.
Assuming it is a tank style WH, It will not damage your water heater (WH) if you shut it down properly.  In order to drain your hot water pipes you need to shut down your WH. (Some homes have drain valve on the hot line out of the WH and this will allow you to drain the hot water lines without draining the WH if the supply line in is off.)
The first thing you do turn the WH off. If it is a gas unit there will be a dial or control pad that will have some settings such off, pilot, and the option to set the temperature from low to high.
Next turn off the gas supply at the valve to the WH.  (for those that have an electric unit it will have switch that supplies power).
Next you must close the water supply valve that is on the inlet side of the WH. ( OR, in your case, the main supply line from the street will shut  off all water.)
Once the water, power/gas and controls are off   then it is safe to drain it, You may want to wait for the water to cool down, and the rest of the system.