Add the heavy cream and vanilla extract to a large mixing bowl and beat on high speed with an electric mixer until stiff peaks form when the beater is lifted. This should take about 2 minutes.
Add the cake mix and sweetened condensed milk to the mixing bowl and mix gently until well combined.
Divide the ice cream evenly into three separate bowls. Add yellow food coloring to one and orange to another, leaving one undyed. Stir each color until the food coloring is well incorporated.
Pour the undyed (white) ice cream into an 8x4 metal loaf pan, ensuring it only takes up about 1/3 of the pan (tilt pan at an angle). Repeat with the orange and then the yellow.
Freeze the ice cream for a minimum of four hours for soft ice cream, or overnight for hard, scoopable ice cream.
Notes
To heat treat the cake mix, spread it out onto a baking sheet lined with parchment paper and bake at 350 degrees Fahrenheit for 5 minutes.