Gas Furnace blowing hot air into room, flue is cold
First:  The situation you describe I would consider borderline hazardous if not clearly dangerous. (CO poisoning hazard) Please do not continue to run this appliance under these conditions.

Now to my practical input;  I worked on boiler (we call this type of appliance a boiler) once that displayed this type of problem. The firebox in these things is lined with a fireproof insulation.  One of these blocks of insulation fell from the 'ceiling' of he firebox and dropped onto the top of the heat exchanger.  This blocked the normal flow path of the products of combultion up the vent and they were generously spilling into the room/house.  Once I reattached that block of insualtion back in its rightful place, everything was fine.  However, the boiler required a couple of hours of my time with a long handled wire brush and a shop vac to rid the exchanger of the resulting abundant soot buildup. 

When natural gas does not burn properly, efficiently and completely the result can be and often is the generation of CO gas. This is a dangerous substance when inhaled especially for children and little ones; in can be fatal.  Please, when making decisions regarding the operation of these appliances, always put the health of your family first.