In the song The Other Side MAX & Ty Dolla $ign pronounce the word "ten" as "teen" in the dialogue at the end of the song. Can anyone explain why?
I don't hear "teen". In any case, vowel sounds are often distorted somewhat in singing: see this paper for some detailed analysis. To me, it seems that the word is lengthened somewhat, which could contribute to the use of an altered vowel quality.
There is a known sound change that could theoretically be relevant here (the "pin-pen merger"; usually, speakers who use the same vowel sound in both words use a vowel closer to the vowel that unmerged speakers use in "pin"), but I don't think the pronunciation of this particular word is enough evidence to show that the speaker has this merger. In any case, the pen-pin merged vowel is still distinct from the /i/ vowel that is found in words like "teen".
The pronunciation in the video sounds like [ tʰɪən].
Pin–pen merger: Before nasal consonants (/m/, /n/, and /ŋ/), dress /ɛ/ and kit/ɪ/ are both pronounced like [ɪ~ɪə], making pen and pin homophones
To people whose dialect does not have the merger, or to non-native speakers, [tʰɪən] could certainly be mistaken for [ tʰijn], the common pronunciation of "teen".