Can having an incorrect hair color on my FAA medical certificate get me in trouble?
This probably won't cause you any practical trouble unless you go for a new certificate or rating (the data on your medical needs to match the data on your IACRA form which needs to match your existing certificates - an examiner may refuse to conduct a test if there are discrepancies).

It's something you should correct though: It's presumed your hair color on the medical is your natural hair color, and if you get ramp checked you don't want to have to explain the discrepancy.



Counterintuitively the answer for "How do I fix this?" is buried in the IACRA FAQ:


  Q:    The data on my medical certificate does not match what is in the FAA Airman Registry database, what should I do?  
  
  A:    If it is less than 60 days since the date of issue of your certificate, contact your AME (Aeromedical Examiner) for resolution of this issue. If it is 60 days or more since the date of issue of your certificate, contact the FAA Aerospace Medical Certification Division for resolution of this issue. The contact info is below.  

Aerospace Medical Certification Division, AAM-300 
Federal Aviation Administration Civil Aerospace Medical Institute 
P.O. Box 25082 Oklahoma City, OK 73125
Tel: (405) 954-4821