In a combustion reaction, is carbon used to make reactants burn?
In a combustion reaction, a substance is oxidized; of course carbon can be oxidized, and the most usual oxidant is dioxygen gas,
So, in effect, oxygen gas is used to make the carbon burn:
No, you do not necessarily have to have carbon involved in a combustion reaction at all.
Combustion is basically the reaction of a substance with oxygen, creating heat and light. The process is commonly carried out for the purposes of creating heat and light for our living spaces, so quite often the fuel will be carbon (or carbon based), e.g. oil, wood, coal, natural gas, because these are commonly available substances that can be easily handled for use in the home (especially things such as coal and wood anyway). However, combustion does not have to include carbon.
For example, you can combust sulphur to create sulphur dioxide, or metals like magnesium (magnesium oxide is formed).
So to answer the question: what makes reactants burn? The answer is simply: energy + oxygen. Supply sufficient heat energy in an oxygenated environment and a fuel will combust (carbon or not).