What is responsible for the smell of incomplete hydrocarbon burning and what sensor can detect it?
In analytical chemistry, sensors refer to analyte specific devices, e.g., glucose sensors, pH sensor etc. Smell on the other, generated by a burning complex mixtures such as fuels can rarely be associated with one compound. People have spent their life (e.g. Alan Marshall) studying the constituents of fuels. There are literally thousands and thousands of compounds.

However, if you were in a research lab investigating odors, you would be using something called as a olfactory gas chromatography technique. There is a chromatography column which separates all the gaseous/volatile components and a scientist puts a nose at the end of the column and records the "response". Later you can identify that particular odor by a mass spectrometer.

In short, there is no sensor per se. You need a instrumentation which would fit in a kitchen sized area to determine what causes that particular odor after burning an extremely complicated mixture such as fuel oil.

Here is how it looks like: