No, you don't. The original Red Velvet Cake relied on the chemical reaction of the buttermilk, vinegar, and cocoa powder for coloring.  If you choose not to add food coloring your cake with be more of a mahogany color. Opt for the food color to get that classic red cake. I prefer using liquid food coloring as it mixes well with the cocoa powder. You do have options for natural food coloring such as using beet root powder. I have not used this but adding 1/4 cup to the cocoa powder should result in a nice red cake color.