Your example uses the negative. Let's start with a positive version and go from there.
Remember to bring the washing powder!
This is an order (the imperative), and the verbs "to remember" and "to forget" imply the future in an order:
Remember to bring the washing powder [when you come over later]!
So the response must be about the future as well. Two common ways of talking about the future are "will" and "be going to".
Okay, I will [remember] [when I come over later]!
Okay, I'm (I am) going to [remember] [when I come over later]!
It's not:
*Okay, I remember [right now]!
The negative in English requires do-support for most verbs, including "to remember" and "to forget". It's not natural to say:
*Forget not to bring the washing powder!
Instead we say:
Don't (do not) forget to bring the washing powder!
We still have to reply using the future, so it's:
Okay, I will not [forget] [when I come over later]!
Okay, I'm (I am) not going to [forget] [when I come over later]!
If you use "I don't" it means you're talking about habitual behaviour.
Don't drive through any red lights!
It's okay, I don't [drive through red lights] [anyway]!