Can I die from bursitis if I ignore it?

No BUT. Bursitis is a localized inflammatory process that causes pain. It is usually self-limiting and even if you ignore it, it may resolve in time.
Not kill you. A bursa is a fluid filled sca that muscles - tendons rub across. They cushion these as they move across bone sites. They can get inflammed with calcium depostis, gout deposits, arthritis deposits but also sometimes infection. Infections rarely can kill you but they need identification and treatment. Untreated infections can be deadly.