my dishwasher manual doesn't say to run tap. in fact it says "The dishwasher can be connected to a hot water supply for further economies. If the water is heated by for example, solar panels, this would be energy efficient. However, if your water is heated by electricity we would recommend connection to cold water. "
i also went to american lg website and checked manual of cheapest dishwasher that they have. it doesn't instruct to run water either. it says that if water is not hot enough cycle will run longer.
anticipating comments that LG is not american enough, i went to GE and checked manual of cheapest (349) dishwasher. it doesn't instruct to run tap either. it does say just like LG that if water not hot enough, cycles that use hot water will take more time (because water needs to be heated)