"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.)
It would be nice to link to the source that you quoted all of that from: italki. Notice that none of the examples are using いいです. I do not think that the original poster was talking about the grammatical structure that we are talking about in this sentence.
Yuki at YesJapan says:
NAIDE (MO) II? (or ...NAKUTE (MO) II?) is a casual version of this sentence pattern.
The polite way of saying "is it OK if I don't ~?" is ...NAIDE (MO) II DESU KA? or ...NAKUTE (MO) II DESU KA? ...NAIDE is a verb NAI-form DE.
...NAKUTE is a verb NAI-form - I KUTE.
Here are some examples:
おんせんに いかないで(も) いいですか。(ONSEN NI IKANAIDE (MO) II DESU KA? = Is it OK if I don't go to an ONSEN?)
しゅくだいを しなくて(も) いいですか。(SHUKUDAI O SHINAKUTE (MO) II DESU KA? = Is it OK if I don't do the homework?)
かんじで かかなくて(も) いいですか。(KANJI DE KAKANAKUTE (MO) II DESU KA? = Is it OK if I don't write in kanji?)
これ、たべなくて(も) いい？(KORE TABENAKUTE (MO) II? = Is it OK if I don't eat this?)
あした、パーティーにいかなくて(も) いい？(ASHITA PAATII NI IKANAKUTE (MO) II? = Is it OK if I don't go to the party tomorrow?)
おかあさんに いわないで(も) いい？(OKAASAN NI IWANAIDE (MO) II? = Is it OK if I don't tell it to mom?)
It seems some verbs use the naide version to be more polite, but nakute is also correct.
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.
Firstly, the subject in the English sentence is "you," not "today," so in the Japanese sentence, the subject should be あなた, which is usually not written out (omitted).
In the sample Japanese sentence, the topic is 今日. There is usually no topic in English ("As for today" is seldom written out). So the topic particle は is optional in the Japanese sentence.