Can seashells burn?

I don't think so. A combustion reaction between CaCO₃ (the most common compound in seashells) and O₂ in the air does not produce anything. However, with heat, CaCO₃ does decompose into CaO and CO₂, so I guess the answer depends on what exactly you mean by burning.

No, not unless you keep them in extremely high heat for an extended period of time, because they are made primarily of calcium.

If burning is understood to mean a reaction with oxygen (one type of oxidation reaction), then the answer is clearly no. Under the right conditions (approximately 840 degrees C, according to this site) it decomposes as redsoxfan325 states.

The balanced equation for this decomposition reaction does not involve any oxygen, and it will occur even if the calcium carbonate has been heated in a non-oxygenated atmosphere. Decomposition is a different type of chemical reaction from burning.

And this isn't even an oxidation-reduction reaction, as the oxidation numbers of all elements remain unchanged in the reaction.

https://www.researchgate.net/post/Does_anybody_know_how_to_reduce_decomposition_temperature_of_Calcium_Carbonate

Here is a video where a seashell is heated up to 2,500 degress Celsius. It doesn't burn but calcium oxide starts to emit a brilliant white light when heated.

https://www.youtube.com/watch?v=1h97IdbDFYo