I don't know of any dialects that consistently use different phonemes for the first consonant after the vowel in it and its or it's.
In many dialects, the phoneme /t/ is pronounced differently depending on its environment, and this applies to it and its/it's the same as it applies to bit and bits/bit's, pit and pits/pit's, and so on.
In an American English accent, a word-final /t/ that comes after a vowel sound or after /r/ can be lenited, causing it to be voiced and realized as a "flap" or "tap" sound (typically transcribed as [ɾ], although as you mentioned, to English speakers it often sounds like a "d" sound) before a following word that starts with a vowel sound. This can be heard in sentences like Give i[ɾ] a rest!, I felt a bi[ɾ] uncomfortable, He spi[ɾ] a pi[ɾ] on the ground. The /t/ in words ending in /ts/ cannot be realized as a voiced flap/tap. So you would not hear [ɾ] in its, it's, bits, spits, pits. Word-final /t/ is not flapped before any consonant sound, not even approximants like /w/ and /j/, so the /t/ in it is not pronounced as [ɾ] in sentences like It won't be long or Give it your best shot.
Plosives in coda position tend not to be aspirated when they are followed by another obstruent consonant in the same syllable. So words ending in /ts/ would usually not be heard with aspirated [tʰ]. Likewise, words ending in /ps/ or /pt/ (like gaps and apt) are usually not heard with aspirated [pʰ], and words ending in /ks/ or /kt/ (like ax and act) are usually not heard with aspirated [kʰ].
In absolute word-final position, aspiration of voiceless plosives is occasionally heard, perhaps especially for emphasis. E.g. gap and hack can sometimes be heard with [pʰ] or [kʰ]. Likewise, when /t/ is not replaced with [ɾ], it can sometimes be realized as [tʰ] in word-final position. It seems like a possibility (although not the most likely pronunciation for me) at the end of it in a sentence like Forget i[tʰ]!
Glottalization is also possible for word-final voiceless plosives. Glottalized plosives are sometimes transcribed like this: *ga[ʔp], ha[ʔk] or ga[ʔp], ha[ʔk]. For /t/, glottal replacement may occur, where the resulting consonant just sounds like and is transcribed as the glottal stop [ʔ]. So Forget i[ʔ]! is another possibility. Glottal replacement is considered particularly characteristic of accents in the south of England, but I think it happens in American English accents also.
I think glottalization and glottal replacement are also possible in word-final /ts/ clusters, although I'm not sure if there is a difference in frequency.