Can I still play sports if I have bursitis?

See physiatrist . A physiatrist or sports med doc can evaluate you ( in the context of what sports you play or activities you do for work) and can make recommendations to keep you active while fixing the problem.
Depends on the sport. Bursitis in the shoulder can be aggravated by activities that require rotation of the humerus, the upper arm bone. Avoid things like tennis, golf, baseball. If bursitis of the hip is involved then avoid lower extremity activities like cycling, rowing, jogging. A physical therapist can recommend exercises that will strengthen muscles around the involved joint to help decrease the inflammation.
Yes, with caution. Playing with bursitis would be permitted if the benefit outweighs the risk. How severe are the bursitis and the associated symptoms? Does the sport place stress on the area of bursitis? Are you improving or worsening in terms of symptoms and function?