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in the phrase "It's funny", the stress is usually on the first syllable of the adjective:
[ ɪts ˈfʌ ni ]
But what happens when the negation "not" appears?
[ ɪts nɑt ˈfʌ ni ]
I'm quite sure the adjective won't lose stress. Does the adverb "not" need any stress on it? Perhaps a weaker stress than the adjective?
Any suggestion is appreciated.
If we follow the treatment of stress in SPE, when part of a phrase comes to have stress, stresses on other elements are not lost, but merely become subordinated to the new stress. If there is no stress at all, a non-diphthongal vowel will reduce to schwa, so if a vowel fails to reduce to schwa, we can take that as evidence that the vowel does have some stress, though the stress may be at a low level due to having been subordinated to other stressed syllables. (There are complications to this that I won't mention for now, and, of course, many don't accept the SPE stress theory at all.)
So a first question to raise about your examples is: do the first vowels of "not" and "funny" reduce to schwa? My opinion is that they do not. So, given that, in "It's funny" and "It's not funny", both "fun" and "not" are stressed. What the stress levels are, will depend on whether the "not" is taken to be contrastive: "It is funny" versus "It's not funny". If so, the highest stress will be on the "not": "It's n1ot f2unny" (meaning that the most stress is on "n1ot" and a secondary stress is on "f2unny".
If there is no contrast, primary stress will be on "f1unny", following the Nuclear Stress Rule, which give the last primary stress in a phrase (here, the verb phrase) the most prominence. Then, we'll get "It's n2ot f1unny".
I have left the stress of "it's" in the examples out of account. Pronouns are a special case, and I think it might either be stressed or not, here.
In my speech, the vowel of uncontracted "not" always has some stress.
Words with negative meanings in English are always stressed. If you think about this it isn't surprising as English tends to stress those words that are essential to understanding the gist of a sentence (those items that are sometimes loosely termed "content" words). The word not is essential for a proper understanding of the sentence, if you don't hear it you'll understand more or less the opposite of the intended meaning of the sentence.
As a rule of thumb adding a negative word to an English sentence does not affect whether the other words in the sentence are stressed (ignoring words that become contracted with not and other oddities like that). It may however affect the relative prominence of other words in the sentence. In other words it may appear more, or sometimes less prominent than another stressed syllable in the sentence. This just depends on the speaker's delivery of the sentence and any intended nuances.
Excluding emphatic readings like "it's NOT funny", there isn't anything resembling an accent on "not". Because of the vowel quality, the vowel of "not" may seem more prominent than the one in "it's", but change the first word to "that's" and I don't see any difference at all.
EDIT: I've observed that judgments on secondary stress are often messed up when vowel qualities differ (the greater duration of [ɑ] in "it's not funny" may be confused with secondary stress). In "It's funny", the duration difference in the syllables is greatest so judgments that "It's" is unstressed are strong. In "It's not funny", "not" is longer and it's shorter than "It's", which can encourage one to think that "not" is somewhat stressed. By lengthening the first syllable (changing "it's" to "that's"), you reduce the duration difference of the first two syllables, which should convince you that there's no stress on "not" and that any feeling that there is, is a consequence of segmental side-effects.
