At least 80 people died and several others were injured in the blaze.
Siptu has written to Minister Murphy seeking an "urgent meeting" with him.
Siptu Public Administration and Community Division Organiser, John King, said: "I note that in a media interview on Wednesday (28th June), Minister Murphy stated that the Grenfell Tower disaster is going to change how authorities in Ireland deal with fire safety. The Dublin Fire Brigade Chief Fire Officer has also said that the Grenfell incident must be a game changer for the Fire Service operations.
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"It is essential that the views and the experiences of frontline fire fighters are fully taken into account when reviewing the operation of fire safety. Fire fighter representatives have for some time been calling for a programme of risk assessments to be undertaken concerning responses to serious and complex fire emergency incidents.
"Such assessments must then inform what responses are necessary to deal with such incidents in order to ensure the protection of life and property as well as the safety of the fire and emergency personnel responding to them."
Siptu Sector Organiser, Brendan O'Brien, said: "It is deeply regrettable that Minister Murphy failed to respond to our request earlier this month to convene an urgent meeting with Fire Service representatives to discuss these issues.
"Siptu members had hoped that progress on these critical issues would be made through the Fire Services National Oversight and Implementation Group. They still believe that this established process, in which all stakeholders participate, presents the best opportunity for meaningful progress on these matters."
(CD/LM)
