The correct answer should be "It's he or I" rather than the colloquial "It's him or me". Both should be accepted but when Duolingo presents the right answer, it should present the grammatically correct form.
He and I read the newspaper. Correct.
Him and me read the newspaper. Wrong.
The teacher likes him and me. Correct.
The teacher likes he and I. Wrong.
The construction "Him is angry" is wrong. So is "It is him" for the same reason, but it's been used so often that it's become colloquial. Duolingo should recognize this and at the very least only present the correct form.
"Him or I" is wrong. "I" is first person singular subject pronoun, whereas "me" is an object pronoun. In this case the two people being referred to our the subject. In any case they should not be used together for example "He or I" and "him or me" are both acceptable whereas "him or I" and "he or me" are not.
But as in Paulenrique's example above, the implied construction is "choose him or choose me"--& it's incorrect to say "choose he or choose I" (here, "it is" refers to the situation rather than the following pronoun, if that makes sense--the idea being, "it's a case of 'him vs. me'"). Cheespuffs12, I understand what you're saying about it sounding clunky, but in this case (a colloquial ultimatum) "him or me" is correct.