In general, we use made of to describe something that is made of recognisable examples of the thing. The house is made of bricks. The fence is made of planks. The planks are made of wood. We use made from when you can't always recognise the original material because of processing or something. Paper is made from trees. Butter is made from milk. There's also made with, which means it contains the thing. The cake was made with honey rather than sugar. For your example, it might be better to say something like: Don't buy things made with animal products. Don't buy things that contain animal products.

Sad and sad

talking of furs = made of fox