What's "kind of shot that horse in the face" in Easy A?

The horse is a metaphor for the situation (Olive dating). Olive is explaining that her allure is lacking: so much so that its effect on her chances of dating is equivalent to shooting the metaphorical horse in the face.

That is, it 'kills' any chance of her dating, much like a gun would a horse.

The relative violence of the overall metaphor also provides a comical contrast to the remainder of the dialogue. Colloquialisms such as these are frequently used in (particularly British) English to soften the impact of a negative or derisory statement through humourous juxtaposition.

If you shoot a horse in the face, you've got a dead horse and a dead horse can't take you anywhere. Olive is saying that because of her complete lack of allure she had no chance of "the dating horse" taking her anywhere irrespective of her parents' punishment.

In addition to what the other answerers have said, Olive is expressing the pointlessness of her parents’ suggested punishment, similar to the futility of “locking the barn door after the horse is stolen” (which is a common idiom).  Perhaps a better comparison would be forbidding her to drive the car, when the car is broken.