Hi,
My daughter had a little bit of jaundice when she was born, which makes babies sleepy, so she would often fall asleep while nursing. We had to strip her down to a diaper and wipe her with a wet cloth to keep her awake (it was warm out, I wouldn't do it now)- it was a two person job. This was a little extreme, but she needed to eat a lot in order to help her get over the jaundice. After a couple of weeks she would occasionally fall asleep, but it wasn't a chronic problem.
One thing that I read that seemed counter intuitive, if you have bright lights on (which I had been doing to keep myself awake in the middle of the night) it can be too bright for babies so they keep their eyes closed and if their eyes are closed it makes them want to sleep!
Congratulations on your new baby and enjoy it. Get a couple of good books and learn to read while nursing. It helped me to feel like I was doing something interesting for me too (not to say that I don't love nursing, but in the beginning it seems like all you do and it can get a little draining!)