Yellow flames leave soot
Incomplete combustion, and why the colour is a warning.
Yellow flames leave soot because the glow that makes them yellow is coming from tiny particles of unburned carbon passing through the flame, and any flame producing enough of those particles to glow visibly yellow is, by the same measure, not burning its fuel completely.
Where the yellow light comes from
Complete combustion needs enough oxygen reaching every part of the fuel fast enough to react it fully into carbon dioxide and water, leaving nothing solid behind. Wherever that oxygen supply falls short, either because too little air is available overall or because mixing has not brought fuel and oxygen together quickly enough, some of the fuel's carbon never gets the chance to fully react and instead clumps together into tiny solid particles, glowing brightly from the flame's own heat as they drift through it, the same physical process that makes a piece of metal glow when it gets hot enough. That glow is the yellow light seen coming from the flame. Most of those particles do eventually burn up if they stay in a hot enough part of the flame long enough, but some escape unburned, rising as visible smoke and settling, eventually, as the black soot found on the inside of a chimney or the bottom of a pan.
A flame's colour is therefore a direct, visible readout of how completely the combustion inside it is proceeding. Blue means the reaction finished with almost nothing left over. Yellow means some of the fuel's carbon did not finish reacting before the flame's own heat made it glow and go on its way, unburned, into the air.
The same fuel can burn either colour depending entirely on how it is delivered. A candle and a gas hob can burn chemically similar hydrocarbon fuels and produce completely different flames, one glowing yellow with soot and the other burning a clean blue, purely because the hob mixes air into the gas before it ever reaches the flame and the candle leaves the air to find the fuel on its own. The deciding factor is the mixing.
The colour also shifts with how much unburned carbon is present. A flame carrying only a trace of soot glows a pale, barely tinted yellow at its core, while one starved of air for longer produces a richer, more distinctly orange flame and, eventually, visible dark smoke once the particles arrive faster than the flame's heat can burn them away.
The black ring on a cold spoon
Holding a cold plate or spoon briefly over a candle's flame, something most people have done at least once as a small experiment or a party trick, leaves behind a ring of black soot exactly where the flame's yellow region touched the cold metal. The plate did nothing to cause that mark beyond being cold enough to stop the rising carbon particles in their tracks before they had finished burning, catching physical evidence of something that was already happening invisibly in the flame the whole time.
A clean blue flame held under the same cold plate leaves nothing behind, because there was no significant population of unburned carbon particles travelling through it in the first place. The soot on the plate was already being made by the flame, and the cold metal simply caught it in the act. The bottom of a saucepan held too long over a poorly adjusted burner picks up a black smudge for the same reason, while a well-adjusted one never leaves a mark at all.
Why a little soot looks like a lot
Only a small share of a flame's carbon needs to escape unburned before the flame reads as visibly yellow, because glowing solid particles are far brighter than the faint blue light given off by the gas reactions around them, and even a thin haze of them dominates what the eye sees. A flame can therefore look dramatically yellow while still burning the great majority of its fuel completely, which is exactly why colour works as an early warning of incomplete combustion and works poorly as a measurement of how much fuel is being wasted.
A faint yellow tinge appearing at the very tip of an otherwise blue flame is a small, early signal, worth investigating but short of proof that the flame has suddenly become inefficient overall.
A candle is a useful exception, since it is designed to burn yellow deliberately. The glowing carbon particles that would be a warning sign of wasted fuel on a gas hob are, on a candle, the entire point, because the light is the product being sold and the heat is incidental.
A blue burner that turns yellow
Because the colour is a direct, visible signal that needs no instruments to detect, a flame turning yellow that used to burn blue is one of the more useful early warnings available on any burner, worth investigating before it is dismissed as merely a cosmetic change. It usually points to something specific: a partially blocked air inlet, a burner running richer than it was designed for, or a draft interfering with the air supply. All of these are worth fixing both for efficiency and because unburned carbon escaping a flame rarely travels alone, tending to carry other, less visible products of incomplete combustion along with it, carbon monoxide among them. A sooty pan bottom also makes the underlying problem worse over time, since the deposit itself insulates the metal it coats, slowing heat into the food and tempting a cook to turn the burner up further still.