My crown fell out a long time ago, and I feel a lot of throbbing pain there now. What's the worst that can happen?

Hi:From the symptoms you are describing most likely there is nerve damage and infection.  You either need a root canal and another crown or worst case scenario you need to extract the tooth depending on the depth and locatio of the cavity.Best,Dr. Maddahi

The worst that can happen to the tooth is that it needs to be removed. The worst that can happen to you is more severe. An untreated dental infection can lead to abscess or spreading of the infection. There are rare instances where the infection can spread and become life threatening. I would be seen as soon as possible. With your sore throat, if you feel any swelling or tightening of the airway, go to the ER immediately.

Before the invention of anesthetic, dental disease was one of the leading causes of death.  It is important NOT to wait.  It may not be significant, but without a doctor evaluation you just don't know.