In a large saucepan, melt the butter over low heat. Add the mini marshmallows and stir continuously until completely melted and smooth. Remove from heat and stir in the vanilla extract.
In a large mixing bowl, combine the Rice Krispies cereal with the melted marshmallow mixture. Stir until the cereal is evenly coated.
Divide the mixture into three equal portions in separate bowls. Add a few drops of red food coloring to one bowl and mix until the color is evenly distributed. Repeat with blue food coloring for the second bowl, leaving the third bowl plain.
Lightly grease a 9x13-inch baking dish. Press the red Rice Krispie mixture into the bottom of the dish, creating an even layer.
Next, add the plain Rice Krispie mixture on top of the red layer, pressing it down gently. Finally, add the blue Rice Krispie mixture on top and press down to create an even surface.
In a microwave-safe bowl, melt the white chocolate chips in 30-second intervals, stirring in between until smooth. Drizzle the melted chocolate over the top layer of the Rice Krispie treats.
While the chocolate is still warm, sprinkle red, white, and blue sprinkles over the top for a festive touch.
Allow the treats to cool completely at room temperature for about 30 minutes before cutting into squares.