Yes, you can injure your neck or back doing push ups. Make sure your back is straight and plank like, and don`t let it dip as you get tired. Don`t strain your shoulders toward your ears or hold your neck tight. Try to keep a long line from the top of your head to your tailbone.
When I was first doing the 100 push up challenge, I really hurt my neck and back. My chiropractor showed me how to do push ups without straining my spine.