Continuity mistake: Vera fires the gun 3 times, and later tells Justice Wargrave that the gun is empty, meaning that all the cartridges in it would be spent. Even if that were true, there should still be three spent cartridges in the gun. When the judge loads his one bullet into the chamber, he does so in a gun where all five cartridges are unspent.
Deliberate mistake: When the judge shoots himself, the revolver slides across the table, to fake his murder by someone else. But there is no mechanism in place that explains how he does this, making it appear as if he is able to accurately throw/slide the gun after shooting himself. In the original novel, he used a rubber band to achieve the effect, but he is not shown setting it up here.