This video tutorial shows how to create a non-destructive distress, grungy effect in Adobe Illustrator using an opacity mask. You could simply place your distress element over your vector image and cut it from your artwork. But the problem with doing it this way is that it will make it difficult to edit your original artwork in future.
So the better way to do this is to place your distress graphic over your artwork, but create an opacity mask instead. This way leaves your artwork intact and also allows you to easily remove or edit the distress elements in future.