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I'm wondering about the "short i" vowel that exists commonly in American English in words like "India, tick, lid". Note, this vowel seems a bit unstable in English and gets merged with others in the American South and other places. IPA represents it as ⟨ɪ⟩. It's the near-close front unrounded vowel.

I have been told this vowel does not exist in French, but I swear I've heard it before. If I know it doesn't exist it will help me never accidentally say it.

Edit: Thanks to input, now that I now it's the near-close front unrounded vowel, I was able to look it up on wikipedia, and it says that it does not exist in French except allophonically in Quebec French. So I believe the answer to this one is "no, it does not exist, so I should endeavor to never say it". As an American I have a fierce tendency to use it in e.g. "s'il vous plâit".

The vowel which is normally written with the letter I in French is a close front rounded vowel, API symbol [i]. Its realization is fairly stable across French speakers, at least in Europe.¹ French does not have a different [ɪ] (near-close front unrounded vowel) sound.

This vowel exists in English, but only as a long vowel [i:]. However, to French ears, the short vowel [ɪ] sounds so similar that many French speakers pronounce it as a short [i]. If French people pay attention, they'll perceive [ɪ] as between [i] and [e] (and I think that's how it's taught in French schools).

Conversely, English speakers might pronounce the letter I as [ɪ] instead of [i] when it's unstressed. French speakers might not even notice. Note that French does not have phonemic (i.e. meaningful) variations on stress or length: stress comes solely from the word and sentence structure, and unstressed speech sounds boring but does not hurt comprehension.

You may find the Wikipedia articles on English and French helpful. It's difficult for a layman to really understand what all these variations are, but most phonemes have an audio sample, and it at least gives a sense of what is (near-)identical or similar across languages.

¹ _{A small minority pronounce it in a more rounded way, sort of halfway towards [u], but this is nonstandard.}

To round out the other answers, some varieties of Canadian French have a series of lax vowels not present in other varieties of French.

These vowels are

• /y/, which can be realized [y] as in standard French, or [ʏ]
  


• /u/, which can be realized [u] as in standard French, or [ʊ]
  


• /i/, which can be realized [i] as in standard French, or [ɪ] ← The one you're asking about
  
  
  • The exact realization varies by dialect. I've heard [ɨ] and even a diphthong like [ɪ^j].
  
  

These extra vowels are allophones of the phonemes /i/, /y/, and /u/. The lax variants appear in closed syllables, i.e. any syllable where the coda is filled. Hence, you will hear paradigms like this:

citer [siˈte] ~ cite [sɪt]

lutter [lyˈte] ~ lutte [lʏt]

router [ʁuˈte] ~ route [ʁʊt]

This pattern is actually an extension of standard French's distribution of tense and lax vowels, including the pairs [o] ~ [ɔ], [e] ~ [ɛ], and [ø] ~ [œ]. Standard French has gaps in the system.

Canadian French is also distinguished by a few words with a final consonant absent from other varieties of French, including frette (< frais ?), litte (< lit), icitte (< ici), and toute where you'd expect tout. It's interesting to speculate (as I did in undergrad) about a symbiosis between this phenomenon and the extra lax vowels.

There is no so called short i (/ɪ/) in French. The French i sound is short, of length approximately that of i in "pick", but of the very same quality of English or american double e as in "weed", or better "wheat" but shorter still than in this latter (the /i:/ sound is shorter in "wheat" than in "weed").

No, French does not have the /I/ sound as in kit and bit. The French grapheme i (written letter) as in petite, is pronounced like /i:/. There are other realizations of the i grapheme as well.

That's why French speakers (and Spanish and Portuguese speakers too) cannot make the difference (unless taught or have a really good ear) between the minimal pairs like ship/sheep or bit/beat/beet or chip/cheap.

Whoever said French has that sound was misinformed. The /I/ sound is not "unstable" in English. Say minute, that letter i and that u are both /I/. And the /I/ sound exists in all varieties of English.

"In English, both in Received Pronunciation and in General American, the IPA phonetic symbol /ɪ/ corresponds to the vowel sound in words like "kit" and "English". It is one of the two vowel sounds we use in English for unstressed syllables, the other one being /ə/.1

In some dictionaries the vowel of KIT is written /i/. There is no confusion as long as the user knows the symbol for /iː/ (the vowel of FLEECE)."

English phonemes

French has the vowel sound /i:/ as in petite. The letter i is pronounced ee in some words as in beet/heat.

Quebec French (French: français québécois; also known as Québécois French or simply Québécois) is the predominant variety of the French language in Canada, in its
formal and informal registers. (Wikipedia, Quebec French
article; see also Canadian French and this answer)

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Tense vowels (/i, y, u/) are realized as their lax ([ɪ, ʏ, ʊ])
equivalents when the vowels are both short (not before /ʁ/, /ʒ/, /z/
and /v/, but the vowel /y/ is pronounced [ʏː] before /ʁ/) and only in
closed syllables. Therefore, the masculine and feminine adjectives
petit 'small' and petite ([p(ø)ti] and [p(ø)tit] in France) are
[p(œ̈)t͡si] and [p(œ̈)t͡sɪt] in Quebec. In some areas, notably Beauce,
Saguenay–Lac-Saint-Jean, and (to a lesser extent) Quebec City and the
surrounding area, even long tense vowels may be laxed. (Wikipedia,
Quebec French phonology article: listen to it.)

Therefore this ⟨ɪ⟩ (near-close front unrounded vowel) exists in French for some contexts with some varieties of the French language. Whether one chooses to use it or not if of no consequence whatsoever and is irrelevant.

Le français québécois, aussi appelé français du Québec ou
simplement québécois, est la variété de la langue
française parlée essentiellement par les francophones du
Québec. (Wikipédia, article Français québécois ; voir aussi
Français canadien et cette réponse.)

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Les voyelles /i/, /y/ et /u/ subissent la règle de relâchement ([ɪ, ʏ,
ʊ]) en syllabe fermée lorsqu'elles sont en fin de mot : « mur » se
prononce [mʏːʁ] mais « emmuré » se prononce [ɑ̃myʁe]), « six » se
prononce [sɪs] mais « système » se prononce [sistɛm], « lune » se
prononce [lʏn] mais « lunatique » se prononce [lunatɪk] et « route »
se prononce [ʁʊt] mais « dérouté » se prononce [deʁute]. (Wikipédia,
article Prononciation du français québécois :
l'entendre.)

Donc ce ⟨ɪ⟩ (voyelle pré-fermée antérieure non arrondie) existe en français dans certains contextes avec certaines variétés du français. Qu'on choisisse ou non de l'employer est absolument sans conséquence et hors propos.