“What happens if you leave a broken tooth untreated?”

Depends. If the tooth is abscessed (infected - typically obvious by X-ray), then leaving it alone is not a good idea because you're now dealing with a systemic infection. If, on the other hand, you've simply broken off a cusp, or a relatively small chunk of tooth, then you're really not in any danger by leaving alone. You do need to see a dentist to verify the extent of the break.

Griffin Cole, DDS NMD

In a few cases, nothing, in some cases sensitivity, and in other cases, a root canal infection requiring treatment or sometimes a deep cavity requiring tooth removal.
I hope this answers your question.

A broken tooth may lead to tissue irritation/infection/abscess and pus drainage, involvement of facial spaces and can even advance to sepsis/Ludwig angina (which is life-threatening). 

It’s best to see a dentist as soon as possible after breaking your tooth. The urgency depends upon the extent of the break. If the break only involves the enamel then leaving it untreated can cause sensitivity & the chipped area could also decay. If the break extends into the dentin then the likelihood of sensitivity is greater. Treatment by the dentist involves bonding composite to repair & restore the fractured area of the tooth. The dentist will also monitor the tooth over time to see if the trauma that caused the broken tooth did in fact damage the blood supply to the tooth. The tooth may turn dark & require a root canal. If the break extends into the pulp (& you see a red area in the middle of the tooth) then you need to see a dentist as soon as possible. The dentist may try a pulp cap, but usually you need a root canal to clean out the pulp & seal the tooth. Then the tooth will require a crown. If you don’t see a dentist & your pulp chamber is exposed, then the infection can cause a severe infection & pain. I don’t want to scare you, but the infection can extend into your brain or into your body (a systemic infection). The results can actually be life-threatening. In summary, it’s best & safest to see a dentist as soon as possible if you break your tooth.
Good luck.

Dr. Fenn

What can happen is infection, pain, more decay in the tooth that leads to pain or infection. You could also have bad breath. All depends on how it's broken. You should have it evaluated by the dentist to see what the options are. It usually is not recommended to leave it alone.

This is a very common question and relatively easy to answer. When a tooth fractures, it needs to be restored with a crown as soon as possible. Although it is an expensive and not pleasant appointment, you actually save time and money by doing it right away. Postponing treatment will only lead to more extensive treatment (root canal therapy, gum surgery or even extraction and implant placement), increasing cost and a number of required appointments.