The patterns of contrast and neutralization
In an open word-final syllable with primary stress, the distinction between FLEECE /iː/ and /ɪ/ is definitely neutralized in both the "General American" and "RP" reference accents (only strong iː is possible*).
In an open word-final syllable that does not have the primary stress of the word, there is a possible two-way contrast of word-final "strong" iː vs. "weak" i~ɪ. The contrast can be interpreted as a stress contrast instead of as a contrast in the identity of the vowel. Compare the OED transcription of manatee ("Brit. /ˌmanəˈtiː/, /ˈmanətiː/, U.S. /ˈmænəˌti/") to that of humanity ("Brit. /hjʊˈmanᵻti/, /hjuːˈmanᵻti/, U.S. /hjuˈmænədi/").
As far as I know, there is no context where any group of speakers has a three-way phonemic contrast between /i:/, /ɪ/ and /i/. It is not standard to analyze them as three distinct phonemes.
Tertiary stress analyses vs. "weak vowel" analyses
There are different views about how to analyze English stress. Some analyses consider it possible for syllables after the primary-stressed syllable to have some kind of stress: this may be called "secondary stress", or it may be called "tertiary stress" to differentiate it from the phenomenon of secondary stress on syllables before the primary-stressed syllable. (In transcriptions, this kind of stress is either unmarked, or marked with the secondary-stress symbol "ˌ"; I don't know of any special tertiary-stress marker.) Other analyses treat all syllables that come after the primary-stressed syllable as lacking any kind of stress.
The phonetician John Wells falls into the latter camp, and wrote a blog "irritating hamburgers" where he explains the reasons:
native speakers tend to perceive the penultimate syllable [of "irritating"], teɪt, as being more strongly ‘stressed’ than the final syllable ɪŋ. But what they want to call ‘stress’ is arguably no more than a way of saying that the vowel is one of the strong ones. Actual rhythmic beats following the main word stress accent are all pretty optional, which is why the British tradition is not to show any secondary stress in words like this: ˈɪrɪteɪtɪŋ, not *ˈɪrɪˌteɪtɪŋ. The alternative tradition, usually followed in the States and (for example) Japan, is to recognize a secondary stress on the penultimate, írritàting.
Rather than using stress marks on syllables after the primary-stressed syllable, Wells's transcriptions make use of different vowel symbols to differentiate "strong" and "weak" vowels: the symbol i is used for a "weak" vowel (which would be identified as being in a fully unstressed syllable in an analysis that recognizes contrastive stress on syllables after the primary-stressed syllable) and the symbol iː is used for a "strong" vowel (which could be analyzed as having secondary/tertiary stress).
"neither /ɪ/ nor /i:/, but /i/" might not be the best way to put it...
The transcription is not meant to imply the existence of a three-way contrast between /i:/, /i/ and /ɪ/, and since there is no three-way contrast, it's not obvious that it's correct to say that i is "neither /ɪ/ nor /i:/". In "happY again" (2012 June 7), Wells says
The symbol i does not mean “neither long nor short”. It means that RP traditionally has lax ɪ in these positions, but that many speakers nowadays use a tense vowel like iː. Therefore the EFL learner may use one or the other indifferently in these cases, because it does not make any difference whether the vowel is tense or lax. See further the discussion in LPD under "Neutralization" (p. 539 in the third edition).
In LPD I use the symbol i in those cases where some people have a tense vowel in place of the traditional RP lax vowel: namely, in weak positions that are
(a) word-final, as happy, coffee, valley,
(b) prevocalic, as various, euphoria,
(c) in the unstressed prefixes be-, de-, pre-, re- and certain word-like combining forms such as poly-.
i: and i are far from the only pairs of "weak" and "strong" vowels (or vowel symbols) that we would have to use in an analysis (or transcription) that doesn't refer to stress or rhythm
For some reason, it's not as commonly discussed, but Wells uses the same analysis for the back high vowel(s), using three symbols uː, ʊ and u and to represent a "strong" tense vowel, a "lax" vowel, and a "weak" vowel that is supposed to be able to be realized as either tense or lax depending on one's accent. (Wells writes "All the above applies, mutatis mutandis, to u (“sitUation”").
It's harder to find examples of word-final u mentioned in the literature, but continue seems like it could be an example. There is some discussion in a 2007, January 3 blog post by Jack Windsor Lewis ("Blog 011", "HappYland Revisited etc"):
What has become a problem for pronunciation lexicographers is that, having given recognition to the weak /i/ of happy etc, they could hardly refuse to recognise the parallel weak /u/ of thankyou for which / `θæŋkju:/ if fully strong must surely sound unnaturally deliberate delivery or a bit of a regionalism. But so far they've done so very grudgingly:both the Longman Pronunciation Dictionary and the (Jones, Roach et al.) English Pronouncing Dictionary give /-u/ for the name of the letter w only in second position. Yet ˈdouble u and ˈw make a quite feasible minimal pair of sorts. For the noun thank(-)you LPD gives only /-u:/ while EPD gives only /-u/. They both agree on recognising (only as a subvariant) /-u/ as a possibility in continue. Only LPD shows /-u/ as possible for value. There are scores if not hundreds of other words that are obviously entitled to equivalent treatment including eg Andrew, avenue, cuckoo, curfew, guru, Hebrew, issue, impromptu, jujitsu, Lulu, menu, nephew, rescue, residue, sinew, statue, tissue,venue, virtue, Zulu etc. And that's only to cite open syllables but there's no suggestion that the allowed examples can't have /-u-/ in their plurals and past inflections. And if continued why not eg prelude? And also /-u-/ in bedroom, costume, granule, vacuum, volume and so on? Similarly, it was an unremarked but completely justified, indeed happy, innovation in the first edition of LPD to have broken away from tradition by showing medial weak /-u-/ where the next sound was a vowel instead of the Jones and Gimson practice of always showing medial /-ʊ-/ in such situations as eg in a word like graduate etc. However, though some speakers may feel /ʊ/ to be their target value, as clearly Wells does, when the medial sound in question precedes a consonant, I certainly don't identify it as my target as regards most such words and I wonder how many others share that feeling about words like accusation, acupuncture, adjutant, aluminium, ammunition, cellulite, erudite, communist, computation, educate, immunise, impudent, manufacture, resumé, tabulate and many others. With some of these I vacillate between /-u-/ and schwa.
(This and another relevant blog post by Windsor Lewis are mentioned in another, older blog post by Wells: "The happY vowel", 2008 May 7.)
We run into a similar phenomenon with the supposed distinction between "strong" NURSE and "weak" lettER (which can be transcribed as /ɝ/ and /ɚ/ or as /ɜr/ and /ər/) in American English: if we transcribe secondary and tertiary stress, we can use /ər/ for both of these sounds. So the OED transcriptions for "dramaturge" are as follows: " Brit. /ˈdramətəːdʒ/, /ˈdrɑːmətəːdʒ/, U.S. /ˈdrɑməˌtərdʒ/, /ˈdræməˌtərdʒ/".
We may also see a distinction between "strong" GOAT and "weak" grottO. It's hard to find good examples of unstressed word-final "strong" GOAT, but for me, I think an (at least possible) example is "veto", which I don't (always) pronounce with a flap t. I do pronounce "Plato" with a flap t, but Greg Lee's answer to my question "What are the historical justifications for first-syllable stress in the word “orthoepy”?" indicates that some speakers have a "strong" vowel in the second syllable of "Plato" (or equivalently, we could say that they have a secondary stress on the second syllable).
*Smoothing may affect stressed vowels before other vowels
In some accents of British English there apparently exists a process of "smoothing" that can change even stressed iː and uː to ɪ and ʊ when another vowel follows. In "listen once more", Wells says that in the RP of 1982 or thereabouts
”Smoothing” may make a diphthong monophthongal when before another vowel: throwing /ˈθrəʊɪŋ/ [θrəɪŋ], diabolical [daə-]; /iː/ and /uː/ may become [ɪ, ʊ] before a vowel: two o’clock [ˈtʊəˈklɒk].
The "phonemic" analysis of "happy" i
There isn't really a clear answer to the question of whether "i" should be regarded as an "allophone of /iː/" or an "allophone of /ɪ/". I believe the very concept of the phoneme is still disputed, so there are big theoretical questions that you'd have to resolve to get a single correct answer to specific questions like this.
My impression is that Wells would be more likely to identify the "happy" vowel (in his own pronunciation) with /ɪ/ than with /iː/, but he describes his own pronunciation (with a quality something like [ɪ]) as "old-fashioned", and I don't think it's obvious that speakers who use [i] should be treated as having the same phonological system.
Windsor Lewis says (in the blog post cited above):
As to the phonological analysis involved, having always considered myself to be aiming at an [i] quality for the happy vowel, I see no point in not assigning my target vowel to my /iː/ phoneme while fully acknowledging that I quite often in less than deliberate speech produce the realisation [ɪ] in various segmental and rhythmic contexts. I am equally capable of producing [u] when my target is /əʊ/ in eg follow or [e] for the latter vowel in essay. So are plenty of other people but it's pretty unusual for any lexicographer to advocate a neutralisation symbol for such words. I do, again like most people, feel there's a phonological difference between eg the final vowels of pedigree and cavalry but the best practical-cum-theoretical way to view them seems to me to be as involving a phonological distinction that resides in a rhythmical contrast without excluding either of them from the phoneme /iː/. A distinct weak-vowel system if you like.