Yes. All airlines will require a birth certificate, because in order for a child to be able to sit in your lap during flight instead of having a paid ticket and occupying a seat, he or she will have to be under the age 2. This is an FAA regulation. It doesn't matter if they ask for it or not, you are still required to carry it. If they do ask for it, and you don't have it, you'll be required to purchase a ticket for the child. I know this because I work for a major airline.
The Delta Airlines do required a birth certificate for infants and toddlers to fly on a parents lap. The birth certificate will offer a valid proof to make sure that they meet the airline's limited age requirement for a lap child.