A dirty air filter slows the airflow, increasing energy bills because it takes longer for the HVAC system to condition your home.
Dust enters your system and coats the blower motor, shortening its lifetime.
When the evaporator coil used for exchanging heat for the cooling system is covered with dust, it loses efficiency. If you run it too long with a dirty coil, you risk burning out the compressor, the most expensive part inside an air conditioner.
A dirty heat exchanger used in combustion furnaces loses efficiency and may develop cracks that emit CO into your home’s air. Replacing a heat exchanger is costly both in terms of part replacement and labor. Unless your system is under warranty, it’s almost always cheaper to replace the furnace.
Section 3.4.1 Air Filters

Preventing some of the wear on your HVAC system is as simple as checking and changing the air filters. Skipping filter changes and running it with a clogged filter is one of the most common causes of system breakdowns and high energy costs. The DOE reports that you can save from 5 to 15 percent on conditioning bills by changing the filter when it’s dirty.
 
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Book Excerpt from air filter section

Section 3.4.2 Types of Air Filters

A Quick Rundown
Consult the owner’s manual or your HVAC contractor to learn the best filter for your system.
Make sure it fits perfectly.
Avoid reusable filters.
Before opting for a better filter, be sure that it will work with your air handler. It must be the right size for your system and density.

HVAC filters carry MERV (minimum efficiency reporting value) ratings that indicate the smallest particles the filter will trap. The scale runs from 0 to 16 and higher numbers indicate the filter removes smaller particles.

Using a filter with too high a MERV rating can harm your system by slowing the air flowing through it. Check with the owner’s manual for your HVAC or locate its specifications online or from an HVAC contractor. Some air filters don’t state their MERV ratings, but instead go by quality descriptions like “good, better, or best.” If your system is fairly new, it may be able to handle a top-of-the line filter, but it’s always wise to consult the system’s literature before making the change.

The filter must be the perfect fit for your system. If it doesn’t, unfiltered air can enter the system. A filter that doesn’t fit well can also be pulled into the equipment or the ductwork, creating problems, some of which can be serious and expensive to repair.

If you can’t find the right size, you’ll have to have them custom-made, either by a local HVAC contractor or an online company. A contractor may be able to fabricate the “rack” that holds the filter, and you can purchase the filter media to insert inside it.
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