Residents of Apollo House, an empty office block in Dublin city centre occupied and refitted as a shelter by the Home Sweet Home campaign before Christmas, led the petition.
The building was once used by the civil service and is now controlled by receivers after property loans were taken over by the state’s Nama.
Irish Housing Network activist Tommy Gavin said: “The Minister for Finance is failing in his duty to the people of Ireland by not exercising his powers under the Nama act to direct Nama to make proper and adequate housing stock immediately available to local authorities around Ireland.
“The housing crisis is bigger than Apollo House.”
Home Sweet Home secured more than 4,000 signatures of support for the petition.
There were more than 1,200 families homeless in the last week of November, according to official figures, including 6,985 people staying in emergency accommodation during December, with 2,549 of them children.
The open letter to Mr Noonan calls on the Government to identify properties, loans, land and other assets which are controlled by Nama and could be used to house homeless people.
It said Mr Noonan should intervene directly and order Nama to prioritise its social mandate, including the delivery of social, public or affordable housing within a set time. The campaigners also said that Apollo House or an alternative vacant property should be made available immediately for sheltered accommodation for homeless people for at least six months.
A resident of Apollo House, whose name was given as Mitzalo, said: “The Government is forcing us into hostels where the people next to you are using drugs and smoking.
“We have great facilities at Apollo thanks to the generosity of Irish people. We have our own rooms.
“We have everything we need. They should keep this place open until the Government builds social housing that they’ve failed to build time and time again.”
The occupation has won support from musicians Hozier and Glen Hansard, filmmaker Jim Sheridan, and actress Saoirse Ronan among many others.
The campaigners are facing eviction next week after the High Court gave them until January 11 to leave the building, which has been empty since 2015.
Home Sweet Home said Apollo House has provided shelter to 40 people and helped another 205 people access shelter over the Christmas period.
It said Nama-controlled buildings should not be used as “honey pots for private development”, but as public housing and high-quality accommodation to end the homelessness crisis.