Is bursitis contagious?

No. Bursitis is "inflammation of a bursa" - in medical terms anything that has "-itis" means " inflammation of..."... Ie: tendonitis, epicondylitis, arthritis..... A bursa is a fluid filled sac that sits where a tendon is attaching to a bone.... And usually overuse of that tendon and its associated muscle can aggravate the bursa - causing bursitis, so nothing at all contagious....
No. Bursitis is not contagious. Usually, it is an inflammation of the bursa (joint pad) caused often by strain or overuse. In some cases, an infection can develop but this is not common and in that case, the infection is usually confined to the joint and so is not contagious.