Warm water works fine. I usually use room temp water. It really doesn't impact the incubator temp that much. At room temp I've been told it takes 15mins for the middle of the egg to cool and people have had eggs survive power outages with the incubator down around 70-80 for short periods of time. I don't think using cool water could keep the incubator temp that low for the length of time required to cause damage to the eggs. Using water from the tap you also shouldn't be able to get water hot enough to over heat the eggs. Although I think it would be easier to use too hot of water than too cold.
Yeah, room temp water is fine. About the only reason you would need to add warm or hot is if you need the humidity to come up quickly, like when hand turning, or when removing an early hatch from the hatcher.