The heat can cause damage. My aunt recently saw I have an old celluloid dresser set. She told me the pieces could burst into flames and suggested I get rid of them. Celluloid is a semi-synthetic plastic invented in the 19th century. It will not burst into flames without help.
Celluloid can melt and burn easily, however.
Never place your dresser set near a fireplace candle or light source including a light bulb or strong sunlight. That is why decanters and vases often discolor. Drain them upside down.