I am assuming this is a fairly regular or consistent symptom for you. It is so difficult to determine what is muscle pain, inflammation or strain, and what could be caused by the heart. In general, heaviness in the chest should be checked. You say you feel as if you have a 'weight' on your chest. I wonder if it is difficult to draw a deep breath? The fact that fingertips are numb could indicate muscle pull or pinched nerve from the neck or chest area. This can also cause discomfort in the chest wall. I have had a lot of experience with muscular discomfort in my chest. I also have GERD, which causes heartburn. And I also have had real heart issues. So I must pay close attention to chest and arm symptoms, heartburn, heaviness, tightness...that said, most of the time these are not caused by the heart for me, but by muscles or excess stomach acid.
As always, I recommend talking to your physician as soon as possible about these symptoms. A lot will depend on your age, physical condition, and medical history. Good luck!