Reminder: I'm not a doctor and please don't consider this medical advice.
The general understanding of carbon monoxide gas is that it is completely undetectable by the human body. We don't smell it and it doesn't necessarily lead to shortness of breath. What usually happens is that carbon monoxide poisoning results in confusion and headache, followed by death. Indeed, the presence of carbon monoxide poisoning is often mistaken as the flu: body aches, headache, nausea and fatigue.
In severe cases, the victim can sink into a coma and have heart trouble, which could include fluid in the lungs.
I couldn't find any studies related specifically to carbon monoxide exposure and the type of inflammation that could cause a sore throat. Instead, there is research showing inflammation in the lungs and respiratory tract (airways) as a result of air pollution -- carbon monoxide as well as a bunch of other pollutants that come from incomplete combustion.
Carbon monoxide release from a furnace is usually from incomplete combustion and that should certainly include some other pollutants. Even worse would be smoke leaking from a wood or pellet stove into your home.
So, the presence of carbon monoxide in the air of a home could mean other gases and particles may be there as well. Those other gases and particles could possibly cause irritation and inflammation in the throat.
Or, it could be a cold.
Anyone concerned about carbon monoxide poisoning should visit the doctor.