Do animals need water to keep themselves clean?

Self-grooming, using areas separate from their nest, roost or burrow as a toilet; eating only their naturally-selected diet, a lot of exercise (mainly in foraging for food).

Some birds indulge in water- or dust- baths, the latter to remove parasites.

Some animals receive symbiotic services such as the fish that will nibble away dead skin from surrounding healthy tissue, the tick-birds on large African Bush animals and the curious little eyelash-mite that lives on us and other mammals. Although these mites can cause problems of their own, depending I think on species of mite and/or host.

There is an enormous range of scavengers that help keep the habitats fairly clean too.