Gas fired hot water tank not connected to transfer switch

I had a transfer switch installed for my home back up generator last year. Haven’t had a blackout so haven’t run it beyond a one hour whole home test. For some reason I doubled checked to see if the hot water tank is on the transfer switch. It is not. On top of the tank is a separate exhaust blower plugged into its own socket . I have to check but the hot water tank because it is gas doesn’t need to be wired into the electrical panel. Am I correct? However, the exhaust is not on the panel nor on the transfer switch. So will this allow the hot water tank to run in an electrical shut down and create a carbon monoxide build up because the exhaust fan will not be getting power? Thanks