Can we determine the presence of carbon by simply heating or charring the substance?
Not all carbon-containing chemical char. For example, sodium carbonate, $\ce{Na2CO3}$ just produces $\ce{CO2}$. Sodium carbonate might be viewed as inorganic, though.

An organic chemical that behaves in a similar way is hexamethylenetetramine, or hexamine, $\ce{(CH2)6N4}$. On heating, it decomposes primarily into formaldehyde, $\ce{H-CHO}$, and ammonia, $\ce{NH3}$, though there are traces of other gases such as $\ce{HCN}$ and $\ce{CO}$. For that reason, hexamine is used as a smokeless fuel (no char left behind, as the saying goes).