'do(es)n't not need to' = '[behøver/trenger] ikke å'
I've accepted 'doesn't have to', as you wouldn't be able to tell the difference unless you knew the intonation of the speaker. 'must not' would emphasize 'ikke', while 'does not have to' would emphasize 'må'.
It's most likely that this sentence would be said to forbid the giving of food to animals, because if you allowed people to feed the animals, you'd probably express yourself in a clearer way.
I get the what is meant now by the sentence but to me it give as correct sentence the 'doesn't have to' which didn't really implied to me that you're not allowed to. So i needed to go to the comments to see why it's wrong, because the correction of 'doesn't have to' is not a clear enough difference of why 'doesn't need to' is not correct.
Technically "doesn't have to" could work, but that would be uncommon in this case. There is no reason to accept uncommon meanings, as it might just cause confusion for listeners. If Duolingo is accepting uncommon ways of saying things, people might start using phrases wrong.