it sounds to me like the heater control valve is either stuck or bad. I think that this valve is vacum operated. check for vacum at switching valve. this valve is located under the hood, it has lines coming from radiator to it.
Second Answer
I don't believe that the Wrangler has a heater valve. This is generally used on high dollar automobiles. The purpose is to cut off flow of the coolant through the heater core under control from the heater control knobs.
More likely, the problem is either the core not getting hot(blocked) or air not being properly diverted through the core. A simple check on the coolant flow is to feel both heater hoses going into the firewall from a cold start. The two hoses should warm up at about the same rate and get uncomfortably hot. If it appears that the core is functioning properly, the problem is likely with the blend door.
The blend door is under electronic control and the actuator motor is located on the bottom of the plenum box on the passenger side. Remove the motor and examine the axle for breakage or cracking. This is a common failure on the Wrangler and Cherokee. You should be able to manually turn the axle and switch from heat to AC. If it appears that the door is bad, there is an aftermarket kit from HeaterTreater that will address and fix the problem at a fraction of the dealer cost.
The front expansion valve is clogged.
heater core could be rusty and old or line to heater core could be plogged most likely old heater core
It mite not be a fuse have you checked the vacume hoses that come from intake manifold this can cause the fan not to blow.
Heater core may be defective
Vent bypasses heater core when open,it is for outside air only...I know this because I just replaced heater core.
Probably need to add antifreeze. Check the Radiator not the overflow tank. If you are low on antifreeze it will not circulate through the heater core.
Leak in heater core. Replace core.
heater core possibly plugged and leaking
It doesn't really blow it out it just warms the cold air.
heater will not blow on a 1996 ford contour
if the heater does not blow out warm air change the setting.
Defective blower motor Resistor Pack. Replace it.