If we treat my child's puncture wound of the toe will it be gone, or do I need to keep worrying?

Probably not. If properly evaluated, treated and managed, the wound should heal uneventfully.
When to worry. Most puncture wounds heal rapidly with first aid treatment but there can be exceptions. If you have reason to believe that a foreign body is lodged in the puncture wound, then it usually has to be removed for healing to occur. If it is not at the surface, then a doctor must be consulted. Also medical issues such as diabetes or chemotherapy make us worry more!
"we" means who? Treat it how? On the other hand, puncture wounds always go away-- it's the bactria left inside that could've a problem-- especially tetnus. Only a dr can set if treatment is needed.