The heater on my 2004 mercury mountaineer does not work. When turning the dial from ac to heat, cold air blows out while a sound like a door opening and shutting is heard.

It sounds like the motor that opens the heat blend door is malfunctioning... They sell them for most makes and models at your local part store... Also a Haynes or Chilton manual to give you specific direction for your vehicle. Also try removing both heater hoses from your heater core at the firewall and now compressed air through the heater core... It may be restricted

Yes, or the resistor.

number of possibilites for no heat.

1. Your engine thermostat is broken and stuck in the on possistion. In the winter months, if its stuck in the open posistion, the vehicle may never get warm enough for the heater to blow hot air (i had this problem once, 13 degrees outside and even tho we had been driving for an hour, not a single bit of warm air... fixed instantly with new thermostat) this is a cheap repair

2. you have a clogged/broken or something has happened to your heater core, it is located in the dash somewhere and hot coolant flows into it. This is not terribly expensive, but labor cost may be a bit high, sometimes there not easy to get to

3. and least likely is that your heater door wont open. Does your a/c still work fine? if yes, if you leave the ac on, and turn it from cold to hot, does it stay exactly same temp or cold? or slightly warmer? if it stays the exact same tempature of cold and you cant tell that anytinh has changed, the door that closes the ac and opens to heater could be stuck/broken, etc. or the cable or electric wire controlling it could be bad..

that concludes 3 reasons you could be without heat, as far as your electrical goes, I would make sure all your fuses are in good, otherwise, who knows, electrical in cars is a whole nother ball game

Your ac blend door is broken or the actuator that opens and closes the door is broken. The ac blend door moves up and down to direct air flow through the heater core(for heating) or evaporator core(for cooling).

When the door mechanism breaks, the blend door falls down and you will only have ac but no heating.

You can buy a ac blend door kit for your explorer, they cost about $120 on ebay or other places.
search for Explorer ac blend door kit.

The duct is under the the dash on the drivers side. And you will have to cut open the duct to install the replacement door.

Heres a link to a tutorial of how to do the job.

http://www.heatertreater.net/Ford%20Explorer%2002-08.html

2004 mercury mountaineer.heat problem
When turning ac knob to heat clicking sound like door opening and closing repeatedly sounds and air blows cold instead of hot.

If I understand your problem, what you actually hear is a "clicking" sound. If I am correct, your unit only blows out cold air. If so, your problem is the "Heating & AC blend door actuator. It is a small plastic box that contains plastic gears. The unit receives an electric signal to change positions, but the broken cogs prevent the unit from operating properly. A quick check to determine if this is your problem, turn the heater on. Adjust the temperature from heat to air. At some point the noise will stop.

Everything that is heat and cooling in a vehicle is controlled by vacuum from the engine. It appears to me that your blend door vacuum line to the temp control switch is either off or cracked/broken and not making things work the way you want/expect. It's the vacuum friend.

could be the blend door motor or the door could be broken these are common

The clicking noise is a defective air temp control blend door motor, this is all to common and issue, to repair the defective part the dash must be removed.

You can't even reach the door without pulling the dash assembly out, then u might as well fix the problem,