The reason it's relatively common to say a movie has a "happy ending" is because movies are often specifically designed to be like that. A large percentage of the audience want and expect things to end on a positive note, just as when we were children we liked our bedtime stories to end with "and they lived happily ever after".
The two main problems with looking for an "antonym" are, firstly, that there aren't so many cases where it could be applied. Secondly, alternative ending types may be quite varied, so there's no generally applicable term.
Endings can be tragic, inconclusive, disturbing, etc., but in general these terms would apply to the whole movie, not just the ending. Sometimes the ending is described as a cliff-hanger (perhaps the director plans a follow-up movie, and wants the audience to be curious about how things are going to be resolved).