When your heater smells like formaldehyde it is often a problem with the heat exchanger, such as a crack which is a serious issue because having a cracked heat exchanger can allow for harmful gases, such as carbon monoxide to leak into the house.
Also furnace components such as the blower motor or electrical wiring can cause a burning smell or chemical smell if they are malfunctioning or overheating.
And sometimes the smell of formaldehyde can come from other household products such as flooring, insulation, furniture etc.
When your heater is giving off a burning smell it could be dust or the newness of the material that the heater is made of.
Dust in electric space heaters and furnace ducts can also cause the heaters and vents to give off a smell of dust burning.
Dirt and debris can accumulate in heater systems and lead to a burning smell.
The material if it's only dust or other debris will burn off and create a temporary burning smell and is often harmless and will go away within a few hours.
Having a clogged or dirty air filter in your furnace for example can restrict airflow and cause the heating system to work harder which can lead to overheating and a burning smell as well as potentially causing damage to the blower motor.
A musty smell coming from your heater or furnace could mean a few things, like a clogged drain, a dirty humidifier filter, or mold growing in your heating system.
Mold can be bad for the health of all occupants, including pets.
If you think there's mold, remove the mold or contact a mold remediation professional to clean it up.
The most common cause of a weird smell from a heater is either dust or mold and dust is the common culprit but you should check the heater out to ensure that it is safe to use.
If you turn your furnace on for example in the winter after it has sat for awhile then it can result in the dust and dirt burning off in the furnace and furnace vents.