Revealing mistake: As Kirk turns to run on the castle steps, an iron bar barricade appears, blocking his path. When it does, watch the castle wall, and the barricade. It seems to slide just a few inches to line up correctly in the shot, showing poor editing.
Continuity mistake: After swiping at Trelane to score a hit, which Trelane avoids easily, Kirk throws the sword away, which hits a bush as it falls. When Trelane calls up another sword, the bush suddenly stops moving.
Revealing mistake: Just before Kirk finds himself in Trelane's "courtroom", look behind Kirk on the bridge above Spock's work station. The viewscreen with the planet on it is actually a piece of paper that's coming away from the wall.