My daughter has a Day and Night furnace that is probably 15 years old. She says the burner ignites when it calls for heat but the fan never shuts off, even after the flame has turned off. Does this furnace have an in-line thermostat or a relay that could be the problem? If so, is it something that I can replace for them?

Most likely has a fan limit control that is sticking or as gone bad.

http://www.drillspot.com/products/47076/White_Rodgers_3F01-131_Snap_Disc_Fan_Control_Switch