I know that the verb changes from 忘れる(to forget) to 忘れないで (don´t forget) but I can´t understand the いて at the end. At first, I thought it comes from the casual imperative form of いる as this post explained but it must have a verb in a て-form before to have this "keep...ing" meaning or to make the request.
This is from a song called "Name of Love", the lyrics are as follows:
未来で逢えるなら 忘れないでいて 僕のこと 本当のこと