When Frasier and Niles are exchanging the real presents with the fake ones, the color of the wrapping paper changes between shots.
When Frasier and Niles are swapping the presents and they realize the mistake they've made, the presents that Frasier is holding change multiple times between shots.
On Christmas morning, Martin takes one of the fake presents from under the tree and prepares to open it. Since the fake presents obviously have no gift tags indicating their intended recipients, why would Martin assume that present was for him?