Although the Old Man is supposed to be way out at sea and all alone, at around 33:00 a small boat with a person on it can clearly be seen between the old man's body and his left arm. Apparently the previous wide shot (Old Man in boat) is reused as background for a following close shot of the Old Man.
(at around 30 mins) There's a white fishing line extending upward from the marlin as it jumps out of the water, which is impossible because Spencer Tracy is sitting at water level. As explained in the end titles, this footage was obtained when a man named Gassell caught a record-sized Marlin at the Cabo Blanco Fishing Club in Peru.
When the Old Man harpoons the fish, there is a closeup shot of the harpoon entering the fish's body. You can see a bloodied hole in the skin of the fish before the harpoon enters it, as it appears to be a reversed shot.
When the boy visits the old man the morning after returning with the fish, the clouds as he walks up the path to the house are in a certain position. When he returns later carrying coffee for the old man, the clouds are in the same position. The scene was shot without the coffee and then immediately with the coffee.
The goof item below may give away important plot points.
In surface shots the dead marlin is about as long as the boat, his fin protruding over the boat stern. However at 59:55, when the first shark attacks, underwater shots show a fish smaller than the boat.