Yes and no. It is illegal to open a fire hydrant by yourself. However, anyone over the age of 18 can fill out a request at their local fire station to have a fire hydrant opened. A firefighter from the station will then open and close the hydrant.
There are two reasons the FDNY does not allow civilians to open fire hydrants. The first reason is water waste. When the FDNY opens a hydrant they apply a spray cap to control the outgoing pressure. Without a spray cap, an open hydrant releases as much as 1000 gallons of water per minute. The FDNY claims this is more water than a family of four will use in a year. By contrast, an open hydrant with a spray cap releases only 25 gallons per minute. The second reason is safety. Without using a spray cap to control the pressure of the water, small children could get hurt by the shear pressure, which could cause injury to eyes or perhaps push them into oncoming traffic.