I think it will be a good idea to start with the verbs "나다" and "하다" because both phrases you took as example are negations of them.
First of all, the verb "나다" is an intransive verb meaning it has to go with "은/는/이/가" while "하다" is often used or acts as a transive verb meaning "to do sth" and the same goes for negations of them.
(*sth - something)
There are two alternatives of negation in Korean.
~(은/는/이/가)나다
negation : 나지 않다 or 안 나다.
~(을)하다
negation : 하지 않다 or 안 하다.
so that, the sentence should be "기억이 안나" not "기억이 안해".
기억이 안해 ☞ 기억을 안해("안해" is still a transive verb meaning it always needs the object marker "을" or "를")
*기억이 안나(the negation of 기억이 나) = I can't remember.
*기억을 안해(the negation of 기억을 해) = She/He won't remember sth.
- Additional answers
Okay. Thank you! Hm, but what does 나다 mean? And what about 기억나. Is that the same 나? And could one just drop the particles and say 기억 안해?
☞ the verb 나다 by itself has serveral meanings in it, but here it can mean "something comes out"
Yes, it's the same as "나" as in "기억나" which is more casual without '다'.
Yes, you can drop markers and it will make the sentence more casual.
Here are some more examples regarding "나다"
생각 나다/생각 나(in casual converation)
- to have an idea or to come to mind
기운이 나다 = to cheer up
냄새가 나다 = to smell
화가 나다 = to be annoyed
Oh and also.. 용기가 안나. This was translated to "i have no courage". Can you also say "용기가 없어"? What is the difference?
☞ "I have no courage" can be "용기가 없어" while "용기가 안나" is more like "I don't feel encouraged or confident."
^________^