Both sentences have bits left out.

Read #1 as "He loves you more than I (do)." That is, he loves you more than I love you.

Read #2 as "He loves you more than (he loves) me."

I guess when you use the subject form of the first person singular pronoun (I), you are comparing that subject (I) with the only other subject pronoun in the sentence (he). When you use the object version of the first person singular pronoun (me) you are comparing it with the only other object in the sentence (you). Does that help?