In short sentences like these :
"What to say and what not to" or "what to say and what to not"
and "What to buy and what to not" which one is correct?
These sentences are examples of ellipsis, where words from the second part of a sentence are omitted because they duplicate a word in the first part of the sentence. If you put back the missing word, you get
What to say and what not to [say].
What to say and what to not [say]
This Ngram graph shows that what not to say occurs frequently, and what to not say does not occur at all.
The first sentence is easily the best choice.
In other contexts, to not say is used, but it is much less common than not to say. See this NGram graph for an example where say is followed by an object.
Many people feel that it is wrong to use a split infinitive, and this may explain the preference for the not to say form.
The same is true if the verb is buy.
You can actually apply further ellipsis and shorten it to what to say and not say. As the second to has been omitted, technically there is no split infinitive. Here is an example, and there are many others like it:
What to Say and Not Say When Giving References and Recommendations - *How to say the right thing every time (Robert D Ramsay 2008)