I have been studying about Russian phonology and its phonemes and I wanted to ask a question: are there any minimal pairs between [ɨ] and [i] or are they just allophonic variations of the phoneme /i/?
The term "minimal pair" means "pair of words distinguished only by the selection of a single phone". As far as I know there is a single minimal pair, икать "to pronounce unstressed е (je) as и (i), the Standard Russian pronunciation, rather than as е (je)" versus ыкать "to make the sound of the letter ы". Otherwise, there are no minimal pairs, because the vowel distinction simultaneously correlates with the preceding consonant ([i] iff preceding C is palatalized). There is a chicken-egg question there, so it is still possible that [ɨ] is an allophone of /i/ appearing after a non-palatalized consonant. Minimal pairs are based on distribution of phones in the surface transcription, and the surface transcription is supposed to reflect the phonetic facts, not the phonological analysis based on those facts (that would be obviously circular). Apart from that verb pair, there are non-Slavic place names with initial Ы, but there would not be any minimal pairs involving such words.