They can weaken wood structures. Like Norway rats, house mice can also spread many diseases. House mice cost homeowners and farmers millions of dollars in damages every year. When the farmer harvests a field, the house mice move to another place. If they cannot find food, many will die.
Young females stop having young when food is scarce.
In most of the world, the population of house mice rises and falls in a constant cycle. There have been many mass migrations of house mice. They bit people and ran over them. Usually they are very timid and run from people. Luckily the cold winters of northern North America prevent such a 'plague' of house mice from happening in northern North America.