In any shot where Moby Dick's tail and hindquarters are the only part visible, it is clear that the model is only being raised and lowered to make the flukes slap the surface, and is not moving in any direction at all.
When Fedallah harpoons Moby Dick, he throws in two harpoons close together on the whale's right flank. As Moby begins to pull the boat through the water the harpoons are in his left flank. As he dives, just prior to the harpoon lines breaking, the harpoons are back in his right flank.
When Ahab goes to pound the gold coin to the mast, it suddenly has a nail-sized hole in the center, allowing Ahab to easily nail it to the wood. Previously, it had been a solid gold coin.
As Ishmael enters the church (around 25:30), the congregation is singing the hymn "Eternal Father, Strong To Save", a song which was not written until 1860 or 61.