As breathing is an essential process for life, any symptoms regarding your lungs are a concern. I would recommend seeing your primary care physician
who could further discuss this with you as well as examine you to make sure there is no serious or treatable process.
As for what could be causing this feeling--there are multiple possible explanations. Some include (1) Problems with the airways. In general, anything that narrows the airways can cause a sensation of heaviness or tightness. Asthma
is a common cause. COPD or chronic obstructive pulmonary disease is another common cause--especially in smokers. These can be diagnosed by your doctor
--sometimes requiring pulmonary function testing. (2) Lung space disease--the alveoli are the small air sacs in the lungs where gas exchange occurs. If these are filled then people can have problems breathing. Things that can fill the alveoli include pus (from an infection), blood (from lung hemorrhage) or fluid (like from heart problems). The alveoli can also collapse--known as atelectasis. Your doctor can help diagnose these conditions--sometimes needing a x-ray. (3) Anxiety can cause the heavy sensation. (4) Heart disease can cause the lungs to feel heavy. There are many other causes, but these are some of the more common.
I would strongly recommend you see your doctor. Good luck.