What can happen if tip-toe walking is ignored and left untreated in an otherwise healthy child?

Usually nothing: Tip toe walking is very common in toddlers and even some children. It is because of the tightness in their heel tendon ( achilles). Walking slowly stretches the heel cords and rarely do you need to intervene medically. Some recommend wearing boots because they can't tip toe in boots. I say be patient and wait. If older than 6-7 years you may want to see your pediatrician.

Gait changes: This is generally a transient gait pattern in otherwise normal and healthy children. Sometimes, this even manifests itself because a new sibling has come along. This will generally resolve with time. However, if it become habitual and if ignored entirely the most likely occurrence is continued and habitual abnormality of the gait with a tightened (contracted) achilles tendon.