If it's a really deep puncture wound, it could become infected. Deep punctures can be especially dangerous because they carry dirt and germs deep below the surface.
I'd say just keep an eye on it. If it gets all sore and the area around it turns red, you might consider going to a doctor and getting antibiotics. But it's unlikely that it will be a problem.
Another thing. Deep punctures (maybe more than, like, 1/4" deep) take longer to heal than just a surface scratch. The skin will heal over it first, because our body's defenses try to close up a wound quickly. But the wound beneath the skin is still healing. So even after the wound seems gone, if you rub it or move it in certain ways, you'll still feel it beneath the skin, either with little twinges of pain, or as a hard little lump beneath the skin, for a while longer. This is normal.