Answer:
Can you die? Yes. Having an overactive thyroid gland can cause a lot of strain on the heart. When your heart is going so fast trying to keep up, eventually the heart will fail. It's not a hard thing to treat, and it's not something that will kill you quickly. It would take years for it to kill you.
However, if you neglect your medicine, attaint a serious burn/injury or take a reaction to a different medicine you can have a Thyroid Storm, which causes vomiting, a comatose state and eventually death.