- My sister and her boyfriend have not been going out together for a long time.
- My sister and her boyfriend have not gone out together for a long time.
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Both are correct. They mean different things.
They stopped dating each other a long time ago (to be going out = to be dating each other).
They are, quite possibly, still 'dating' but they have not actually been out together (on a date) for a long time. (to go out = (er...) to go out!)