Do school district/municipal government retirees contribute to the cost of healthcare? 

Local government employees are entitled to benefits (health, dental, life insurance premiums, etc.) when they retire based on the contractual provisions of a collective bargaining agreement.  Since healthcare contributions are not part of a pension benefit, they are not constitutionally guaranteed and therefore can be adjusted for retirees and current employees as long as they are adjusted at the same rate.

 As has been identified above, the Taylor Law and Triborough Amendment make it difficult for local government to negotiate better rates thus raising the cost to taxpayers for both retired and active employee benefits.

When a school district/municipal government’s rates increase, do the retirees pay a portion of that increase? 

Retirees do not contribute to changes in the pension system contribution rate, since those terms are defined by the pension tier system, but retirees may contribute to healthcare rate increases, depending on the collective bargaining agreement.