For me, there is a minor stress on "Let" and a major stress on "Fin".
Student or Learner
Hi, the four syllable phrase "Let me finish" phonetically looks like: [ lɛt mi fɪn ɪʃ ]. When someone interrupts me, and I told that person: "Let me finish" I think there is primary stress on the first syllable of finish like this: [ lɛt mi ˈfɪn ɪʃ ]. I'm not sure about the verb "let" though, do we need a bit lower stress on "let" when the phrase is pronounced? or we don't need any stress on "let" at all?
I agree with Mike.
I'm not a teacher. I speak American English. I've tutored writing at the University of Southern Maine and have done a good deal of copy editing and writing, occasionally for publication.