Both frowning and smiling burn calories. However, they each only burn a tiny fraction of a calorie in every instance. Both frowning and smiling require muscle movement, and any muscle movement requires the use of caloric energy.
It is impossible to know the exact amount of calories burned with these activities, but one study has found that laughing burns roughly 50 calories per 10 to 15 minutes, or about 5 calories to a little over 1 minute of laughter. Since the typical smile lasts only a few seconds and engages the same (but fewer) muscles than laughter, it is safe to presume that each smile burns only a small fraction of a calorie.