Ms Mallon was speaking after the SDLP met with Minister Charlie Flanagan about the Irish Government response to the referendum and their duty to people in the North this week.
She said: “The Irish Government now has a critical role to play in safeguarding the rights and entitlements of people in the North as citizens of the European Union.
“As co-guarantors of the Good Friday Agreement, the Irish Government has a duty to uphold the principle of consent and the people of the North made it very clear last Thursday – we do not consent to being forced out of Europe.
“Given the stated will of the people of Northern Ireland, it’s critical that we accelerate the process of North/South integration and cooperation.
“One early way of advancing that project is to establish passport offices in Derry and Belfast to meet the surge in demand for Irish passports as people here seek to protect their European citizenship and entitlements.
“The Executive must provide resources to ensure that happens immediately.
“We impressed on Minister Flanagan the SDLP’s commitment to pursuing every legal and parliamentary mechanism to ensure that Northern Ireland stays within the European union.
“We also underlined the absolute need to prevent the re-emergence of a physical border across this island. Any new frontier between Britain and Europe should be around the island of Ireland, not across it.
“The SDLP will continue to build a coalition of voices that will respect the decision made by people here and robustly defend Northern Ireland’s position in Europe.”