What are the potential consequences of high bilirubin levels in the blood?

Depends. This depends on the cause, level, & type of elevated bilirubin (conjugated vs unconjugated). There are many causes of high bilirubin levels, some of which are relatively benign, and some that may be serious. In some cases, high bilirubin may cause yellowing of the skin/eyes or increased risk for gallstones. In newborns, a significantly high bilirubin is dangerous due to the risk of brain damage.
Casuse. The consequences depend on the cause of elevation. Almost any viral infection can cause transient elevation. A benign disorder called Gilbert disease can have slightly elevated levels. Other causes are hepatitis, drug reaction, hemolytic anemia, biliary obstruction etc.