Generally speaking heat helps quite a bit. Nearly all chemical actions are more active at higher temperatures and so would aid the detergent action in removing unwanted substances. It also causes fats to soften and melt permitting greater penetration of the water/detergent mix. From personal experience it is definitely easier and much quicker to wash dishes in hot water.
I do note, however, that some washing powders (for washing clothes) are said to be designed for washing cold. Although this may be also a way of saving energy, the detergents may not be so eco-friendly and a main reason is because of the lower risk of leeching colours from fabrics.