Is it safe to get a navel piercing or other piercings while on the birth control pill?

It is very safe. Their will be no problems If you already have your navel pierced and become pregnant it is safe to keep the piercing in if you're pregnant. There are maternity retainers that flex with the growth of your belly.

Any piercing can become infected, navel piercings have a higher chance of infection than most others, because of the vicinity that it is located at since most people neglect to keep the area clean. Being on Birth Control pills won't change the chance of infection, or make an infection more or less harmful.