I've recently moved into a 3 story town house and have ran into a problem heating my house. I have one HVAC system with a thermostat on the second and third floor. The thermostat on the third floor only controls the third floor and only when the 2nd floor thermostat is off. The second floor thermostat is the primary one and controls the heating and cooling for the house. The problem arises when I am trying to heat the whole house...if I put the main thermostat on 70 it'll be 90 degrees on the 3rd floor. Unfortunately for me that is where my bedroom is located and I literally have to turn off the heat to the rest of the house and only run the 3 floor thermostat at night which makes for a cold morning when you wake up and go downstairs. If people spend the night...well you can see how it's a problem. Is there an issue with my damper system? I'm not opposed to having an HVAC guy come out but I don't wanna pay someone to come out just to tell me how to work it (if indeed it is working properly) Hopefully this made since!!
It sounds like the original installation owners cheaped out on the system. Much depends on the layout of your house. Either way, it seems you would at least want two different units, one to provide for two floors and a second unit to provide for the third floor. With one unit, you're sending all the heat up to the third floor or all the cool air to the first. I would find a reputable HVAC guy to come advise you on what to do.
OP, is this all standard forced air? I assume so. I don't think it's a quick fix -- you need a professional to come tweak your system. With any luck, it's not just a sloppy architecture problem.
You can close all your 3rd floor vents, so all heat has to come up from the lower floors. Probably not efficient.
A cheap solution is to put a gas heater on the lowest floor. Something like a procom natural gas heater.