Less severe sinus infections will usually go away on their own without any medical treatment and usually nothing bad will happen.
However in rare cases for more severe sinus infections the more severe sinus infection if left untreated could cause things such as meningitis, infection of bones and brain abscess.
If the sinus infection gets worse then you need to get treatment for the sinus infection but if it's not very severe you don't need medical treatment.
But still getting medical treatment for any sinus infection is a good idea so that you don't have to suffer from the sinus infection.
Some people have died from severe sinus infections so it's best to take antibiotics for the sinus infection and not take any chances.