Untreated hyperthyroidism can lead to serious complications, mainly related to the heart. When you have hyperthyroidism, your body is, in a way, running on overdrive all the time, and that can greatly affect your heart.
Some possible heart-related complications of uncontrolled hyperthyroidism are:
Arrhythmia (abnormal heart beat, such as atrial fibrillation)
Cardiac dilation (increase in the size of the heart cavities, which actually thins the heart muscle) and congestive heart failure
Sudden cardiac arrest
Hypertension (high blood pressure)
If you don't treat hyperthyroidism, you also run the risk of developing osteoporosis. You can gradually lose bone mineral density because uncontrolled hyperthyroidism can cause your body to pull calcium and phosphate out of the bones and to excrete too much calcium and phosphorous (through the urine and stool).
You need calcium and phosphorous to maintain healthy bones, so if your bones aren't absorbing enough those minerals or losing them at an increased rate, they can become less dense. This can also make your body temporarily hungrier for calcium after thyroid surgery. Eventually, you may develop osteoporosis—meaning that your bones aren't as strong as they should be and making you prone to fractures.
If you suspect some of the symptoms of hyperthyroidism are affecting you, talk to your doctor to minimize the chance of complications from untreated hyperthyroidism.