How is the relationship between the reader and the story affected by movies that are released prior to the completion of a book series?

For starters, it might help to think about some of the differences that we experience when viewing a single movie made from a single book:

• Appearance/attractiveness of characters. Especially for those characters to whom we're attracted, the change of blond hair to brown, or dark skin to light can be jarring. This is always going to happen because film creators are limited by the set of actors who actually exist, whereas of course characters in books are not limited at all. (Eg. Brad Pitt plays the attractive guy, regardless of whether he matches the character's description.)
• Ethnicity of characters. In Western literature, non-white characters are going to be made "ethnically explicit" much more often than white characters. Seeing a character whose race is different than expected on screen might entirely shift the ways in which readers identify with that character. (Is Harry Potter ever explicitly identified as white in the book?)
• Tone of characters or settings. Perhaps you imagined a particular setting as a very dark place, while it's filmed very brightly. Or maybe you see the evil character as darkly serious but the film portrays her as careless and apathetic.
• Emphasizing of different portions of the story. Did anyone else miss Tom Bombadil?

The general effect is one of discordance. What we expect is slanted, mutated or sometimes flat-out untrue.

When we're talking about movies and books being intermingled in the way Harry Potter has, we get two competing effects:

• the discordance of bouncing between multiple interpretations, and
• a synergizing of the content such that the books will provide subtext for unseen portions of the movies, while the movies will gradually shape how we interpret the books.

The quality of the adaptation determines the extent to which we move from the former to the latter over time. If they complement one another, then alternating between book and movie can be an enriching experience. If the films are of bad quality or offend your interpretation, obviously the experience will be less pleasant.

The films overshadowing the books can be another problem, depending on your point of view. This can result in some practical concerns, like:

• They provide a sort of objective interpretation. When friends quote the characters, for instance, they'll likely quote the film version simply because everyone will recognize the reading - even if the line is exactly the same in the book.
• Some readers may ignore remaining books entirely, knowing that the movies will be easier to consume and not too far off.
• The story may not be "done" until the last film, which puts the books in an oddly secondary position in spite of being source material.

Note that similar phenomena can happen outside the book/film context. The toys released in advance of each new Star Wars movie rather ruined some of the fun, for instance.