Does sleeping on the back hurt the baby?

It is a bad idea to force yourself to sleep in a certain position. There is a condition called "vena cava syndrome" where women close to term who lie on their back can lose conscience. This can happen in the operating room during surgery. That's why pregnant women during a cesarean section are slightly tilted, to prevent the uterus from compressing the large vein.
But as long as you are comfortable there no scientific evidence that sleeping on your back will harm the fetus or that sleeping on your side will improve pregnancy outcome. Sleep in the position that's most comfortable for you as long as you are comfortable.

Many women believe that blood flow will be reduced through the placenta if they lie on their backs. The origin of this myth can be found in some valid research originally performed in the 1960s and 1970s, which demonstrated that blood flow can be compromised when a mother is forced to labor lying flat on her back. Compression of the vena cava, a major vessel underlying the uterus, may occur in this position when the mother is in labor. This is why women are encouraged to be on their sides, sitting up or walking when they are in labor. Contractions themselves reduce blood flow to the baby for a certain portion of the peak of the contraction. A healthy term baby can tolerate this stress without difficulty. Laboring on one's back may cause fetal compromise when the baby is already stressed, overdue or preterm, or exposed to infection or a very long labor.

As far as the possibility of compromising blood flow by sleeping on one's back, the mother-to-be in this situation would generally be experiencing inadequate oxygen tensions, and this would be considered a high-risk pregnancy. And, if blood flow was indeed compromised, the mother would feel dizzy and very uncomfortable and she would shift to her side naturally. This again would be highly unusual.