Bursitis -may- eventually go away if the underlying cause heals (most commonly a supraspinatus tear), however this does not always happen. Usually, you need a cortisone injection into your shoulder which will remove inflammation of the bursa (bursitis) and make you pain free. The underlying tear may not heal, but you'll be pain free and they are often of little consequence if they are small tears.
The progression is this
1. See your GP. He'll determine a painful arc and refer you for ultrasound
2. Ultrasound detects bursitis and a cause for the bursitis (usually)
3. You get a cortisone injection and if there is a full thickness tear (unlikely) you'll need surgery. If not, just the injection and perhaps some rehab
Since you say its bursitis, chances are you've already had the ultrasound and your GP should have advised you about the cortisone injection