The reverse question is really the issue. The break doesn't cause immune problems, but what if you already have immune system issues?
If a tooth is chipped /broken to the point that the soft tissue inside the tooth is exposed, then bacteria is entering parts of the tooth that are not meant to be exposed. This is especially important when the break is due to decay. Infection can spread through the tooth and progress beyond the root tip. While anyone could develop an abcess or painful infection, someone who has a depressed immune system could develop a more severe infection. One should not take any infection lightly, but, infections in an immunosuppressed person should certainly be dealt with quickly.
David S. Strassman