This is a very common habit among young infants. While seemingly harmless, if it persists, it can constrict the child's palate (roof of the mouth) and not allow it to develop properly. This will cause the upper (maxillary) arch to be narrower than the lower (mandibular) arch, and can can cause problems with your child's bite.
Teeth position also affects facial features, and can therefore alter the appearance of your child. To illustrate, imagine someone who doesn't have any teeth, their lips and face get sunken in because the teeth aren't there to support them.
Not only will the palate get constricted, but it can cause the front teeth to turn in towards the center and your child may not be able to close his or her front teeth properly. It is very easy for a dentist to identify if the child is thumbsucking, or some other similar habit.
WAYS TO STOP THUMBSUCKING:
Most children will eventually stop on their own, but you should have your child checked regularly by your dentist to see if it is having adverse effects on the mouth.
TREATMENT FOR CONSTRICTED PALATES:
If your child already has a constricted palate, it is possible to correct. Usually an appliance called a palatal expander will be used, which is placed in the upper arch and is periodically tightened to re-expand the palate. This can only be done when a child is still developing, so if you are an adult and have this problem it is likely too late.
Braces, Invisalign, and other orthodontic systems can be used to try to correct this problem as an adult, but sometimes surgery is needed since it is now a skeletal problem and not merely a tooth positioning problem.