We have a Forced air gas furnace with a programmable thermostat. What is happening is that the furnace will turn on when the temp drops, like it is supposed to, and the fan will come on to blow the hot air, but the fan doesn't turn off. It will continue to blow even when the air is cold coming out of the vents.

Would this be something wrong with the furnace itself, or the thermostat.
We rent, and if it is the furnace, it is up to the landlord to fix the furnace, but we installed the programmable thermostat.

Any help would be greatly appreciated.

This is not likely a thermostat problem(unless the thermostat fan setting is set to on instead of auto. I suggest having the owner of the property call an hvac service company to trouble shoot your furnace. Good luck and please let us know how things work out.