1. Fractured Rib
2. Chest Infection
3. Exercise induced Angina
4. Gastroesophageal reflux disease (GERD)
All worth checking ASAP as it is in the correct region for heart related mishaps and the sensation of tightening is definitely something to mention to your GP when seeking an appointment.
Pain that gets worse when you press on that specific area of the chest is probably from the muscles or the bones in the chest and not the heart itself. If you could pinpoint the specific spot where the pain originates with one finger this is a good sign - as heart related pain tends to be difficult to pinpoint.