Is 'not' a modal verb?

"Not" is not a verb. "Not" isn't really the same part of speech as any other word. It is unique. But if you have to classify it for some reason, it seems closest to an adverb. Dictionaries like classifying words by parts of speech, and that's the classification they use: e.g., the American Heritage Dictionary or Oxford Living Dictionaries.

This isn't a well-sourced answer. In general, dictionaries are not a reliable source for information about "parts of speech" from a linguistic point of view, because dictionary categorizations of the "part of speech" of words are often based on simplified or outdated models. I haven't read any linguistic sources that talk about whether not is a verb, but here are some arguments that I can come up with.

Evidence that "not" is not a finite verb form

It's easy to see that in standard English, we cannot replace a finite verb with not:

• She wants to see the movie. (grammatical)

• *She not to see the movie. (ungrammatical)

• She seems happy. (grammatical)

• *She not happy. (ungrammatical)

If not were some kind of verb, it would have to be a non-finite verb form. This makes it unlike the modal auxiliaries (can will may etc.), which characteristically have finite forms, but no non-finite forms.