Why doesn't Calcium Carbonate Burn?
Calcium Carbonate cannot burn. Its already in its highest oxidation state (+2 for Calcium and +4 for Carbon), and cannot be oxidized (combusted) to a higher oxidation state. However, calcium carbonate does decompose on heating to quicklime and carbon dioxide.

$$\ce{CaCO3 -&gt; CaO + CO2}$$

This doesnt give out any flames in the first place.

Now let's talk about how to salvage your project. Why don't you try to heat the calcium carbonate, and show that carbon dioxide is evolved by using a burning match stick, or by using lime water? $\ce{CO2}$ extinguishes the burning flames and forms a milky precipitate with $\ce{Ca(OH)2}$ (limewater). This can be used to show that perhaps egg-shells and marble have carbonate content in them.