Can I use Sunflower oil to make soap? When reading about making glycerine soap, the recipes suggested to melt animal fat and mix it with $\ce{NaOH}$.

Could I use Sunflower oil instead? Or a olive oil for nice flavor? Is heating necessary in that case?
You can use any triglyceride to make soap.  The process is called saponification.  In this process fatty acid molecules are cleaved from glycerol (glycerine).  In order for the reaction to proceed it is heated.  Regardless of the fatty acid source, the soap will probably not taste very good.