This seems related, although slightly different from I question I once posed: Why are Harry's GoF dreams in the third-person?
Throughout Goblet of Fire, Harry's dreams of Voldemort are strikingly different from the ones later in the series. Harry sees them from a third-person view, the scene begins with someone else (Frank Bryce in his first dream, Crouch's owl in the second dream), and Harry can't feel Voldemort's thoughts or emotions. We never get an explanation for the shift, but the best explanation is that it has something to do with Voldemort lacking a full body and full powers.
As for why Harry is riding an owl in the dream... well, it is a dream. Throughout the series, Harry's dreams often shift from normal dreams to glimpses into Voldemort's mind. In his dream where Nagini attacks Mr. Weasley, for example, it originally begins as a silly dream about Cho and the Room of Requirement. It might be that as the glimpse into Voldemort's mind began, Harry was still half-dreaming and compensated so that he was riding the owl, not just seeing it.
I'm also not convinced that Harry is literally riding the owl in the dream. If he was simply viewing the dream from a vantage point slightly above the owl's back, a dreamer would assume he was "riding" the owl.
