"The laundry" is the pile of clothes that needs washing. "Do the laundry" means clean all of those clothes.

Do the washing or do the laundry?

"the washing up" refers to dirty dishes and utensils that need to be cleaned up. "to do the washing up" means to clean those dishes in a sink. "do the laundry" means to wash (and usually dry and iron) dirty clothes.

Do you do laundry in a sentence?

"I like to do the laundry. His wife said she would go do the laundry. They never just wake up, have breakfast, do the laundry. "As my husband says, we don't do the laundry much".