Are there doctors who specialize in medical illnesses that cause worsening of psychiatric disorders?
In general, there are not doctors who specialize in treating medical illnesses that worsen psychiatric disorders.
Some medical problems are notorious for their psychiatric side effects, and experts in these illnesses may be expected to be more knowledgeable about the psychiatric complications associated with the medical problem.
Some examples would be Parkinson's disease, which is frequently associated with depression and hallucinations, and Wilson's disease, which is sometimes associated with psychosis.
A neurologist expert in Parkinson's or Wilson's disease should probably be quite knowledgeable about the psychiatric spects of the neurological condition.
There is also a subspecialty in psychiatry, called psychosomatic medicine or consultation-liaison psychiatry, that deals with understanding and treating psychiatric problems arising in connection with medical disorders.