Ask the doctor if he did a blood screening for arthritis. Have the medications helped ease the pain? Do you do alot of typing or computer work? Is the burning only in your left hand?
You should have had blood test for arthritis and lupus. Please call the doctor and find out what he tested and what the results are.
First I recommend that you see a rheumatologist for examination of your hands and your blood test results.
Ask the doctor how he feels about trying you on prednisone (you need a prescription). It is a steroid and anti-inflammatory, and hopefully it will reduce swelling from the inflammation and and therefore the burning feeling as well. In the meantime you can try taking anti-inflammatory medication purchased over the counter, such as ibuprofen, if it is not contraindicated for you. This is a non steroid medicine that will help reduce inflammation and pain associated with it.
Alot of typing can be responsible for your symptoms. Try wearing braces (from the doc) on your hands for some support and relief, while you type.