Tooth abrasion. Our front teeth are designed to cut into foods--not grind them. Our posterior teeth (molars and premolars) are flatter for this purpose. Chewing nails regularrly can, over time, cause the enamel to wear away and change the look, shape, and size of your front teeth. Sometimes mixing extremely hot sauce with nail polish can help reduce the habit by making the person's nails painful to taste.
Excess wear/fracture. Chewing fingernails can lead to excess wear and chipping on the front teeth. As you misalign the teeth and the edges are compressed together, it can also break any existing fillings, crowns, or bonding that you may have already invested in. In some people this protrusive action of the lower jaw can also affect the tmj. Not a good habit to do.
None. Nail chewing can result in a skin in infection of the fingers. It will not effect the teeth. You may want to try nail polish that is specifically made for nail biting.