"I do not like books that are not interesting."
I think this is mostly due to direction of the statement. ありません is used for things that "exist" like inanimate objects. So your statement is closer to "There is no like for books that are not interesting" which may be considered a bit odd. I have not seen ある and あります used on emotions in this way before, however there are sentences in English such as "There is no love lost there." that are ambiguous and may require ある to make sense in Japanese.
If I understood your question (non-native English), yes it can.
For example, if your friend compliments your new accessory 「かわいい！刺繍（ししゅう）が上手いね！」 you can answer 「ありがとう。でも私が作ったんじゃなくて、買ったんです。」 = "That's so cute! You're so good at embroidery!" "Thanks. But I didn't make it, I bought it." Or if you get strange looks for asking to sleep in the same bed/room with your male friend: 「苗字（みょうじ）は違うけど夫婦（ふうふ）なんです。」"We've got different surnames, but we are husband and wife."
Most likely because 嫌い (きらい) is a な-adjective, not an い-adjective. It's one of the very small number of な-adjectives where the kanji form ends in い, making it easy to forget 嫌い isn't an い-adjective. ^^;
So it's probably rejected in the same way that the following for "I like interesting books" would probably be rejected on Duolingo...
...which is because a な-adjective (like 好き or 嫌い) being used predicatively should be followed by the copula (だ, です,...). At least this is what virtually all resources teaching Japanese require their students to learn for using な-adjectives.
For い-adjectives (高い, 小さい,...) the situation is different. The い at the end of い-adjectives is itself effectively the copula for this class of adjectives. The reason です is often still added after い-adjectives is merely for the sake of adding politeness/formality to the sentence.
嫌いだ or 嫌いです
好きだ or 好きです
高い (with optional です to add politeness/formality)
小さい (with optional です to add politeness/formality)
This is my guess as to the reason your answer was rejected anyway. It might actually be for a different reason? ^^