넣다 is to put / place in.
* 서류를 서랍에 넣었다: I put the papers in the drawer.
놓다 is a widely used word, by itself and as an auxiliary verb.
1) release; let go
* 아이가 손을 놓았다: The baby let go its hand.
* 잡은 새를 놓아 주었다: I let the bird I caught go free.
* 놔 (=놓아)!: Let go!
2) (aux verb) put / ready in certain state (similar to 두다)
* 밥을 미리 해 놨다 (놓았다): I cooked the meal in advance (ready to be eaten later).
* 10 시까지 이 일를 해 놔 (놓아): Get this thing finished by 10 (for me).
* 서류를 서랍에 넣어 놓았다 (=넣어 두었다): (better sentence than the above)
* 기억할 수 있게 잘 봐 놔 (봐 둬): Take a good look so you can remember.
3 put ... on ...
* 지갑을 책상 위에 놓았다: I put the wallet on the desk.
4 free oneself from worries
* 한시름 놓았다: I am relieved now. (A big worry is gone)