The answer to your question has much to do with what you mean by heart and kidney and how you define an animal. 

The smallest metazoans (animals), such as the hydra can survive without either a heart or a kidney, because their functions are not needed.

These organisms are two cell layers thick.  O₂, nutrients and waste products (such as CO₂) diffuse into and out of all of its cells.

As metazoans grow larger, simple diffusion is no longer adequate.  To bring O₂ and nutrients to all of the cells of the body, they must develop a circulatory system.  

To drive cells and fluid through this circulatory system, they need a pump -- a heart.  

The circulatory system also accumulates wastes from the cells of the body.  These wastes have to be removed. 

In vertebrates, the organs that remove these wastes are the lungs, which exchange waste CO₂ for O₂, and the kidneys, which removes non-gaseous wastes.  

Other organisms have analagous organs, such as the nephridium of flatworms and the malphigian tubules of insects.  Like the kidnesy, these are used to collect and remove wastes from the body.