1. The Nanny
The Nanny named Fran didn't end up giving birth on an elevator, but she did get stuck in one in the hospital while in laborbut got out in time. This one makes the top of the list because this also happened in the show's finale, which pretty much drives the point home that if you have to have a baby born on an elevator on your sitcom, you might as well end it right there.
Did I miss any here? Add them in the comments!