Maybe I worry too much, but I want to know how puncture wound of the mouth will affect my child's overall health.?

Varies by severity. Themouth is commonly exposed to trauma and is therefore accustomed and well suited to uneventful healing in instances of minor trauma. In the event of more extensive trauma, meticulous repair will likely minimize deformity or loss of function. Tissue loss or electrical injuries or thermal burns are more likely to result in long term deformity.
Hopefully not at all. However if you have concerns about the severity of the wound, definitely get it checked out. Rarely, severe complications can occur from a puncture wound. If there was any substantial bleeding, if your child appears seriously ill, or if you're not sure, have him evaluated asap. Generally however, small or superficial puncture wounds heal without any specific treatment.
Should Not. Assuming normal wound healing and typical cause of puncture (like a fall, blow to mouth area, etc.), there should be no long-term problems relative your child's injury. Wound healing is a process, so the remaining scar may be the only long-term evidence of the injury.