Why can't "do" be left out of "I do not like apples"?

This is a good question, and I think the answer lies in history. "Do not verb" wasn't always the way things were said. Here's a chart:

_{The use of periphrastic do in Early Modern English negative declaratives: evidence from the Helsinki Corpus}

The "not+V" form was not as popular as the "V+not" form in eModE, but it was a valid way to say things. The form "Do+not+V" came into being after do became used in questions ("Have you any?" vs. "Do you have any?"). "Do+not+V" won out for several reasons:

• English was switching over from SOV to SVO
• It was similar to the existing "Aux+not+V"
• It makes the distinction between object negation and sentence negation clear
  • Example of object negation:

    But she spoke not of a lover only, but of a prince dear to him to whom she spoke
    _{Cited in the aforementioned paper (E3, CEFICT3B, FICTION, SAMPLE 1).}

I also wrote this answer about the Earliest attestation of “does/do/did not + verb”. The information and sources there are also relevant.

You could say "I like not apples".

It would sound rather quaint unless you were using it as irony, for emphasis or for effect. But it would be perfectly idiomatic English.