Recently I was doing a show for my school and massive feedback occurred. Next thing you know the mic stopped working because the batteries didn't work. The batteries were brand new at the beginning of the show. The broken batteries were replaced, and the mic worked again. All other mics worked fine with the new batteries till the end of the show.

My question is, could the feedback have anything to do with the mic not working/batteries being dead.

Yes, most certainly.

In normal use, with no sound into the microphone, the current drawn from the battery is quite low.

With sound into the microphone the current increases.

I both cases the internal circuit is simply an amplifier driving a very low powered transmitter.

When acoustic feedback occurs, the microphone is part of an oscillatory loop and the current taken from the battery increases dramatically.