"넌 영화를 관심 없이 봐", "넌 영화를 관심 없게 봐" => both are fine, and interchangeable.
Both ~게 and ~이 can make an adverb from an adjective. ~게 is the universal one.
~이 and its cousin ~히 appear in more limited cases, usually after ~하다 and some others.
It often depends on the end form of the adjective whether the ~이/~히 adverb may be used or not.
For example, ~없이 is widely used, but ~있이 for 있다 is not.
Often times both ~게 and ~이/~히 exist for an adjective, usually with the same meaning.
There is no simple rule as to when ~이/~히 form may or may not be used, so you must learn it ad hoc.
Examples of different cases:
- 크다(big)->크게, 푸르다(blue)->푸르게, 넓다(wide)->넓게: No ~이 form for these non-하다 adjectives.
- 높다(high)->높게/높이.: Both ~게 and ~이 exist (exceptional).
- 촘촘하다(dense) -> 촘촘히/촘촘하게; 엉성하다(loose) -> 엉성하게 only.
- 딱딱하다(hard) -> 딱딱하게 only.
- 물렁물렁하다(soft, mushy) -> 물렁물렁하게 only.
- 급격하다(abrupt, precipitous) -> 급격하게/급격히.
- 과격하다(violent) -> 과격하게. 과격히 is questionable (used in a very limited way).
- 사랑스럽다(lovely) -> 사랑스럽게 / 사랑스레. (~스러이 is always shortened to ~스레).
As you can see, there are all kinds of different cases, so we can't go by a simple rule about them.