I have a sharp burning pain on my 2 metacarpal finger. 3 weeks ago doc said sprained but no better, so bad I can't make a fist cuz it hurts an swollen?

Have it evaluated. Some causes of pain and swelling can include systemic problems such as hypertension, gout, and problems with the kidneys and localized issues such as arthritis, infection, lymphatic obstruction, blood clots, varicose veins, trauma (fracture, sprain, and tendonitis) and side effects to medication. Have it evaluated by a professional and get the appropriate treatment.
See below. These are symptoms that could be potentially serious. Yous should discuss these with your physician. He/she will thotoughly evaluate you and let you know what's going on.