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There is no fire hazard caused by venting your gas furnace into a masonry chimney. The exhaust from the gas furnace is rather cool compared to oil furnace or fireplace exhaust standards. In fact, many homes have the gas furnace exhaust exit the home at ground level without any chimney at all through the use of special fans and, believe it or not, PVC (plastic) pipe!
There is a real possibility that your chimney may be "oversized". An oversized chimney is one that exceeds the maximum size recommended by the manufacturer of the furnace. A properly sized chimney will have a powerful upward draft... the rush of air in a well-designed chimney that pulls the exhaust gasses up and out! An efficient chimney is small enough to cause a good draft but large enough to accommodate the maximum amount of exhaust gas your furnace can generate. If the chimney is not "drawing" the exhaust adequately or if the liner is damaged, you could conceivably get unhealthy accumulations of carbon monoxide caused by downdrafts pushing exhaust gasses back into your home. Carbon monoxide is an odorless but highly toxic gas is a byproduct of combustion. There is an article on the dangers of CO at https://www.naturalhandyman.com/iip/infsisters/infco.html
With insufficient draft, moisture and other exhaust gasses do not exit the chimney quickly, leading to water condensation in the chimney. The exhaust should heat the chimney walls sufficiently to prevent the moisture in the exhaust from condensing. If not, the moisture can lead to acidic corrosion which can eat at the cement between the sections of masonry chimney liner, eventually leading to holes in the chimney walls and exhaust gas leaking into your home.
Another possible danger of a poorly sized chimney is condensation. Much of the byproduct of burning natural gas or propane is water vapor. If significant moisture condenses on the masonry, there is a possibility of cracking during winter freeze/thaw cycles. This situation is worse if there is exposed brick anywhere inside the chimney, since brick is absorbent and thus also as vulnerable to cracking as cement. Chimneys with terra-cotta liners may have exposed brick if there is an "offset"... a design technique used when the chimney is not totally vertical, but rather shifts sideways. Offsets are common in multiple-flue chimneys that utilize a single vertical chimney for separate fireplace and furnace exhausts.
The only way to know if the chimney is safe and efficient is to have a professional HVAC person inspect it. This pro can also tell you whether or not it is wise or even necessary to install a liner in the chimney, or even if an alternative venting system might be cheaper and safer. Resizing can, of course, be accomplished with a stainless steel liner. If the chimney is otherwise structurally sound, the liner can be installed with little if any masonry modification.
Though chimney lining with stainless steel is surely within the physical abilities of an able handyman, I hesitate to recommend it as a do-it-yourself job due to the potential dangers that could be present if the job is not done properly.
Lucky you! You probably don't ever have to clean it again. The exhaust from gas logs is primarily water vapor, some carbon dioxide and, if combustion is not efficient, some carbon monoxide. These combustion byproducts are not corrosive and leave no residue to speak of so they will have no significant short-term effect on the lining of your chimney.
Of course, it is wise to have an occasional checkup by a pro just to make sure that no cracks have developed in the chimney liner or, if you have a metal chimney, that it has not become rusty. Leakage of the moist exhaust into the walls around the chimney could result in typical moisture related problems ... lifting paint, rotting wood, mildew and insect infestation. This inspection should not cost very much, and should be done at least every 3 to 5 years.
Chimney liners (a fire-proof-or-resistant lining within the brick, stone or cement-block chimney structure) were not always required by building codes, so it is quite possible that your chimney may be unlined. Considering the age of the house, there is also a great possibility that adding a liner may be necessary for your health and safety!
When determining the type of work your chimney may need, one critical question is what type of fuel are you burning... wood, coal, oil or natural gas? If you are burning coal or wood, you should most definitely get the chimney checked out by a pro to be sure it is still safe to use.
If you are using oil or natural gas to heat the water, the situation may be less critical since neither generate as much heat in the exhaust, but only provided that a professional inspection indicates no gas leaks! This is not something that you should do yourself… there is a lot at stake. Both these fuels have another potential danger... namely the carbon monoxide in the exhaust. With natural gas, there is the additional problem of condensation, since much of the exhaust from the burning of natural gas is water vapor. If the chimney is attached to the house and has ANY cracks or breaks in the block, bricks or mortar, carbon monoxide (CO) could possibly leak into your walls. Even if the amounts are not life threatening, exposure to CO can cause subtle and damaging health effects over the long term. As mentioned earlier, the moisture from natural gas combustion leaking into the walls can actually lift interior or exterior paints and even start rot within the walls!
There are two methods of chimney relining currently in use. One requires the insertion of a flexible steel or aluminum tube that acts as a new chimney liner. The second method involves coating the inside of your existing chimney with a special material that seals and restores the existing liner.
You may have another option… alternative venting. If you are using natural gas, for example, you may be allowed by your local building code to vent through to the outside of your house anywhere above ground level… and not use the chimney at all!
In any event you must consult a local chimney inspector (or two or more if you prefer) to get the "skinny" on your chimney's condition. Since different contractors will offer different solutions (and probably different prices) it pays to have at least a couple of opinions. In knowledge is strength!