Permaculture, Sustainable Design, Homesteading, Off-Grid Living, Natural Building, and more

Depending on the livestock and source of water, generally yes.

Irrigation from a water well is almost always safe. Surface water irrigation from a lake or river can sometimes be subject to pollution from toxic algae or wastewater that could lead to disease, but unless there is a known water quality issue, it's fine.

Tougher livestock like cattle, goats and sheep can drink nearly anything. Livestock like chickens and pigs are a little more sensitive to water quality and may not perform as well.