This might shock you, but to us native speakers, 「去{さ}る」 and 「残{のこ}す」 have completely different meanings from each other and, for the life of me, I could not think of a situation where the two verbs can be used interchangeably.
The evil, as usual, is the translation and the bilingual dictionary that would tell you that both mean "to leave".
In the only Japanese-speaking country on earth, however,
「去る」 means "to (physically) go away from a place" and
「残す」 means "to leave something/someone behind"
Hope you could see the huge difference here. Since the two are so different in meaning from each other, you could easily use both in a relatively short sentence. Let me borrow your sentence (by correcting its particle mistakes) to illustrate my point:
『花{はな}の世話{せわ}をするのを忘{わす}れないでください』と言うノートを残した。 "Someone left a note saying 'Please don't forget to take care of the flowers!'"
It is also completely natural to say:
『花の世話をするのを忘れないでください』と言うノートを残し（て）去っていった。 "Someone went away, leaving a note saying 'Please don't forget to take care of the flowers!'"
Finally, your other sentence:
「人間{にんげん}がきらいだから、この世界{せかい}を去りたい。」 "I want to leave this world because I hate people."
is valid for using 「去る」. You want to do the ultimate "going away"; You want to kill yourself.
You can never say 「この世界を残したい」 in this context because "this world" is not your belonging in the first place. You can only 残す the things you own, the persons you love, your accomplishments, etc. Oops, I forgot you hated everyone. ☺
