Assuming you have access to the hinge mechanism. This item will probably be made of cast iron, and the hinge mechanism has rusted solidly.
Care must be exercised to avoid breaking the cast iron as it is a brittle material, and some of its substance will have been consumed into making rust. Thus the cross-section of the material will already have been reduced!
One of the spray lubricants is your best chance, but remember, there is no way to recover the material that has already turned to rust. Give it a good spray, and leave for a few hours for the spray to penetrate.
Try and get a tiny bit of movement. Try some more spray.
Once you have a small movement, then further movement will abrade the rust into a powder. But realize that is what you are doing.
A top damper or cap damper is a metal spring door placed at the top of the chimney with a long metal chain that allows one to open and close the damper from the fire place.
Your chimney damper is open
One can purchase a chimney damper from a variety of stores. Chimney dampers can be purchased from fireplace specialists such as Woodland Direct, or from online stores such as Amazon.
When you use the fireplace, the damper should be open all the way. If not, you probably will not get enough updraft and your house will fill up with smoke...not a good thing. When it is not in use, the damper should be closed...otherwise you will lose a lot of heat up the chimney.
Put on a pair of safety glasses or goggles (you ARE going to get stuff in your face) get a good flashlight, and get a look at the damper. Most are hinged, and will swing UP the chimney. Most have a bracket and support the holds it open or closed. With the light to help you see, move the support so that it is free of the bracket, and try to swing it open (up). If it does not move, you may have foreign material in the chimney (especially if your chimney does not have a cap). If that is the case, you need the services of a chimney sweep.
fireplace
By flue, I believe you mean the damper in the chimney, which has the effect of closing off the flue. Close the damper any time you are not using the fireplace. Remember -heat rises -so heated air will also rise up and out the flue in the winter. In the summer it is also good to keep the flue closed because heat and humidity will activate the smelly properties of the ash and creosote in the firepalce chimney and create a bad odor in the home. In order for a damper to do its job it must seal tightly/ If your existing mechanical damper is broken, missing or in a bad state you can repair it or replace it with a chimney top damper or a chimney balloon. But, open the damper before lighting a a fire, or it will rapidly get quite smokey in the house.
There are many kinds of dampers, depending upon how old the house is and the choice by the masons. Many dampers have a hinge on the rear and a handle that can be pushed or lifted to lift the front and latch it open. Others have a lever designed to move with a knob or chain connected to the damper. Some chimneys have counter-balanced dampers on the top of the chimney that are opened by pulling and latching a chain. A "tilt and slide" damper not only opens with a handle, but also has a movable grating that can be "slid" open and shut using a knob cast into it. If you have a gas log fireplace and can not close your damper by the handle it is very possible that the damper has been disabled or welded open according to the 1991 building code that states that dampers in new gas log fireplaces must be disabled. The only type of damper that is allowable to use in a gas log fireplace is an inflatable damper. If you do not have a gas log fireplace and you cannot open or close your damper it is likely that it has corroded to the point of non-function. The chimney is a very acidy and corrosive environment any metal in it will rust and degrade rapidly over time. Lubricate the damper with a penetrating oil and try to loosen it by hand or call a chimney sweep to come and work on it.
FIREPLACE
The flue is the actual pipe or chimney where the damper is the mechanism that opens and closes the flue
It's important that an indoor fireplace flue damper be opened before igniting anything in a fireplace. The damper looks similar to a trap door and is located at the bottom of the chimney. When it's open, it provides an opening for smoke from the fireplace to escape. If it is closed while a fire is lit, smoke will enter the room rather than leave via the chimney with possible deadly consequences. If nothing is burning in the fireplace, keep the flue damper closed. This saves money on energy by preventing heat or air conditioning loss up through the chimney.
Something is preventing your fireplace from drawing. The damper may be closed, or partially closed. Your chimney may have a blockage, or you may have prevailing winds that push smoke back down the chimney. If you have air flow problems, such as a vent fan that draws air out of the house, it can cause that. Bottom line- if the damper is open, and the flue is not blocked, you need a good fireplace/chimney specialist to examine what you have.