If you find the title of this post confusing—that is a good thing. There really should not be any connection between a conventional gas water heater and an air filter.
As one can see in the diagram below, gas water heaters, unless they are a sealed combustion type heaters, get their combustion air from holes around the bottom of the heater.
This next picture shows the carpet in the room fitted nicely around the base of the heater—completely blocking the air intakes.
This could be considered some sort of air filtering system (bad idea), but it is most likely going to end up starving the heater of adequate combustion air. This could lead to a flame producing large amounts of carbon monoxide.
Obviously, for the heater to function properly and receive proper combustion air, the air intakes must be kept clear. All this carpeting will need to be removed from around the heater.
While it may look better this way—it is very dangerous and once again shows how the work of one trade can negatively impact the work of another trade.
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