なくてもいい is simply the opposite of てもいい, and highlights that it is ‘ok even if (A) is not done’. This usually comes across as being quite polite, and sounds like ‘don’t worry about doing (A)’ in English.
なくてもいい is a combination of the auxiliary verb ない in its conjunction form, with てもいい.

• ミカは悪（わる）くないから謝（あやま）らなくてもいいよ。
  Mika, you are not in the wrong, so you don't have to apologize.
• その手紙（てがみ）はもう届（とど）けなくてもいい。
  It's okay if you don't deliver that letter anymore.

In order to make this phrase sound even more casual, the adverbial particle も may be omitted. This does not change the nuance at all, but will sound a bit more friendly (when used with people that you know well).

• 今日（きょう）は会社（かいしゃ）に来（こ）なくていいよ。
  It is fine if you don't come to the office today.