What should you do with a dead baby pigeon?

DO NOT TOUCH IT. Never touch dead birds.

Birds can have diseases (including some serious viruses like West Nile Virus and Avian Flu and even some worse ones) that they can give you. Even if that was not what killed them, it is still something you can get by touching the body if they had the germs on them. If you must move it, either use a long handled shovel or put on disposable vinyl gloves or, the best method is to use a zip top bag to pick it up and zip it closed around it so you do not touch it and any particles on it are contained. You should also avoid breathing too closely to it so you don't inhale any airborne particles, another reason for closing it tightly in a sealed zip top bag (hold the bag with the top pulled down over your hand picking the bird up through the bag without touching it, then close the bag around it and zip it shut)