ていい and てもいい

Verb ＋ていい and verb + てもいい both mean that doing something is “good” but are usually used to express that it’s ok to do something. They are typically used when asking or giving permission.

ていい

It’s ok to~
Verbて＋いい

Verb て form + いい literally means something like “verb (is) good”. 

It is typically used when asking or giving permission:

• たべていいですか？ — Eat good? – Is it ok to eat it?
• たべていいです。-Eat (is) good. – It’s ok to eat it.
• よんでいいです。– Read (is) good. – It’s ok to read it.

てよくない is grammatically correct, but unused in favor of てはいけない

See also ないでください

てもいい

It’s also ok to (or not to)~
Verbて＋もいい

てもいい, like ていい, means “verb is ok” but てもいい additionally implies that not doing something is also ok. This makes it sound softer and more polite, since you are implying that its ok if the person refuses. 

Asking Permission:

• たべていいですか？ — is it OK to eat it?
• たべてもいいですか？– Is it (also) OK to eat it? (but its OK if I can’t).

Giving Permission:

• よんでいいです。– It’s OK to read it.
• よんでもいいです。– It’s also ok to read it. (and ok to not read it).

See also ても 

なくて(も)いい

It’s ok not to~
Verbない form → なくていい
It’s ok even if you don’t~
Verbない form→ なくてもいい

The negative of ていい is なくていい, and the negative of てもいい is なくてもいい. 
なくて is the negative て form, formed from the ない form of a verb:

• たべない→たべなくて
• およがない→およがなくて

This form expresses whether it is ok to not do something.
てもいい’s nuance changes slightly to ”even if“

• 読まなくていいですか？- Is it ok to not read it?
• 読まなくていいです。Its ok to not read it.
• 読まなくてもいいですか？ Is it also ok if I don’t read it? (Is it ok even if I don’t read it)
• 読まなくてもいいです。 It’s ok even if you don’t read it.

Usage differences

In addition to てもいい being softer and more polite; ていい and てもいい have one more usage difference that occurs because of てもいい implying that not doing something is also ok.

When you want someone to do something,ていい is preferred. 
When you don’t care either way, or you don’t want them to do something, てもいい is preferred. 

• へやをそうじしなくていい – Its ok to not clean your room.
• へやをそうじしなくてもいい – It’s ok even if you don’t clean your room. 
• ひげをそっていいです – It’s good to shave your beard. (It’d be good if)
• ひげをそってもいいです – It’d also be ok to shave your beard. (It’d also be ok if)