Do pleurisy symptoms feel like heartburn?

Generally not. Pleurisy is an inflammation of the lining of the lungs, and many details of this are in prior answers. 'heartburn' is a layman's term of a sensation anywhere from the upper abdomen to the throat of a 'burning' or heat or acid taste. It is generally caused by acid or stomach contents returning into the esophagus, and the pain caused by it is not related to breathing, as is pleurisy.