It's just because English is funny in terms of what you "will." If you said that you will not say good night, you are saying that it won't happen in the future. The point of the sentence is that in the present, you do not say good night. The Danish sentence and the meaning itself are probably just weirdly formed rather than the translation.
The sentence could mean I'm not saying goodnight right now, but also consider people who don't like to say goodbye. They might say "I don't [I never/habitually don't] say goodbye." This person, for whatever reason, doesn't say goodnight (possibly tonight, or possibly ever). The problem is, without knowing more, we have to go with the most general translation
I see what you mean. I believe that the Danish for these must be the same (I know it is in Norwegian). In that case, it's a matter of translator's preference."I will not..." adds something to this sentence that, I believe, isn't there. My approach with DL is to translate rather directly. Your approach to DL, I suspect, is more broad :)