Don't forget to remember me .

Do you remember me?=Remember
Remember what I say.
Opposite:-forget=fail to remember.
forget and remember used together is a style in writing.called Parallelism.
where use of balanced words,expressions is in the sentence:-eg:-
the idea of forget and remember.You can read more to give the impression you want.

"And the love that used to be ..."
-- a song from The Bee Gees that I won't forget to remember (= I'll remember forever).

"Don't forget to do something!" is as simply as a reminder for you to do something.

That is nothing to do with "parallelism in writing"!

Don't forget to do something是叫人記得做某事

而Don't forget to remember me.是叫人記得想起我（重點是要人常常＂想起我＂）

其實中文，我們也會說：「記得掛住我！」

希望幫到你！