I think it depends upon how large the tanks is and how "full" (Full) really is.
A 10 gallon water tank will make virtually no difference. A 100 gallon tank could make a difference.
I tow a travel trailer, not drive a motor home. But I do know when my fresh water tank is full (and it's 45 gallons), I can feel that driving the truck. When it's empty the trailer handles different. There's a certain amount of movement that occurs with the trailer. I can't explain it, but every time I've pull out of port and hit the high seas, I can tell within the first mile if I forgot to open the drain on the fresh water tank. I'll stop first chance I get and let it drain on the road. Within a few miles, the to experience is different as the tank empties out.
About fuel mileage, I really don't know. As far as weight, if you have a 50 gallon tank (and 1 gallon of water weighs approximately 8 pounds), that's 400 pounds of water.
If you carried two passengers with you that weighed 200 pounds each... how much would that affect your fuel mileage? Probably not enough to mess with. So is the case with a full tank of water.
The difference (in my opinion) is how the trailer handles when towing. With a motor home, it depends upon how solid the suspension is, how heavy the motor home is, and how resilient it is to movement inside the coach. If the MH is a behemoth, even 100 gallons of water may be nothing compared to the over all weight of the coach.