"You don't need to take notes today."
When you have a verb+nai, use "なくて" for cause/reason, and use "ないで" for all others.
As toru is a verb and there is no causality being expressed, it is properly "とらないでいいです". If it was an adjective nakute would be correct. As in, "高くなくていいです". とらなくてもいいです is also correct, but would mean, it is alright EVEN IF you do not take it.
Here are the general guidelines:
1) When you have adjectives(i or na)+nai, only "なくて" is correct, whatever it means.
この辞書は大きくなくて、高くない。 This dictionary is not big and not expensive.
安いワインを買いましたが、おいしくなくて捨てました。 I bought cheep wine, but the taste was not good, so I throw it away.
2) When you have noun+ja+nai, only "なくて" is correct, whatever it means.
あの人は彼氏じゃなくて、弟です。 That person is not my boyfriend, but my younger brother.
いい子じゃなくてごめんなさい。 Sorry for not being a good child.
3) When you have a verb+nai, use "なくて" for cause/reason, and use "ないで" for all others
顔を洗わないで学校へ行きました。 I went to school without washing my face.
そんなところに座っていないで、こっちに来れば？ Why don't you stop sitting there and come here(to join us)?
電車が来なくて遅刻しました。 I was late because the train did not come.
Of course there are situations that both なくて and ないで work. 君と会えないで残念だ。Not being able to meet you, I'm sorry. (It's more like situation.) 君と会えなくて残念だ。I'm sorry that I can't meet you.(It's more like reason.)
Don't see how いりません can be used with a phrase (ノートを取るのは要りません is an unnatural sentence). Use the one I provided above ノートを取る必要がないです。
Which is the right kanji for とる here? Based on the hints I picked 撮る, but that was marked wrong. Some people here have suggested 取る. From the dictionary I can't tell. I found there is a cool app called torumemo that uses 撮, but I don't know how it works. Maybe you snap pictures of notes instead of writing them.